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This is a very good point. It's been nearly 40 years since I was in college, but the academic hour is a time-honored tradition going back God knows how long.
I don't think the OP should really feel cheated unless the teacher is mentally absent or otherwise occupied during the actual time she's there. If he gets 50 good solid minutes of instruction and interaction, that's probably what to expect.
This might be differenct in a private home, but he mentioned the word "conservatory," which leads me to think this is a more academic setting, in which case the "academic hour" probably applies.

I would like to say that I respectfully disagree. Granted, my experience in the academic setting is fairly limited, only having attended four years in a preparatory conservatory and then four years at University, but I've never heard of an "academic hour" in a lesson setting. In fact, my experience as a student suggests quite the opposite. MOST of my lessons ran over five, ten, sometimes even fifteen minutes, as professors can afford not to schedule lessons back-to-back-to-back. If we were still working on something, we kept on working. Now, perhaps it took me 75 minutes to learn something that normally could be taught in 50, but that is a separate discussion.

I also think it's false that an hour is "too long" to concentrate fully on a single task. For myself, practice sessions normally go at least twice that, not getting up for more than two or three minutes. Similarly, I've had numerous lessons, again as a student (though it was never the norm), that were scheduled for TWO full hours. Yes, we were both exhausted by the end, but it is certainly possible.

All that said, taking a minute to get a glass of water or arriving five minutes late letting the student "warm up", in my opinion, is not an issue. However, consistently taking 10-15 minutes off the front of an hour every lesson IS an issue.

my lessons are always two hours and both myself and my teacher concentrate for the full time, sometimes longer

Different strokes for different folks. I often grab something to drink in between students, but it takes no longer than the one student learving the studio and the other unpacking their materials and getting organized. There is always a little "dead space" between lessons. That's just the nature of things.

However, 10 minutes is way out of line.

Could it be your institution has scuttled the "academic hour?" I've run into it more than once, at different institutions.

And then your teacher teaches another student for another two hours right on your heels? Followed by another? And, with all due respect, how do you know what's going on in your teacher's mind?

Bear in mind, we're not justifying boorish behavior; if a teacher offers a lesson for an hour, a student has the right to expect the teacher to be present for the hour. We are offering the OP some ideas to consider about what may be happening. We often get just one side of a story here, and the other side is not able to defend itself or clarify their position.

The comments are very insightful and thought provoking, I just thought I would mention and clarify some things.

First a clarification, my instructor gives me the 10-15 minutes of warm-up time, but when she comes back into the room she asks to hear my scales, arpeggios, ect, nonetheless. It is not as though she is listening while I warm-up. In fact I have frequently heard her conversing with other instructors during this time.

Second, I am her first student of the day so she can't be taking a break because of fatigue.

Third, if she is operating on this "academic hour" she has given no mention of it to me. In fact this is the first I have ever heard of such a thing.

I don't know how things are elsewhere in the world, but in the UK the expectation for private music lessons is that the instructor will be fully attentive for the entire period of time for which he or she is paid. OK, we all have to make allowances for calls of nature, but even then a student would not expect the instructor to be absent for more than a minute or two.

In schools the expectation is that an hour's lesson (in any subject) will be a bit shorter than the hour, but I've not seen a single instance (in the UK) of this tradition migrating into the private sector.

Of course one can, in principle, be fully attentive and not be in the same room as the student. But in such a case the student is entitled to an explanation from the instructor how instruction is being carried on.

I appreciate that conventions might be different elsewhere.

PPP is the lesson at a university, or a private music school? Do you know what job she comes from before your 4pm lesson? Maybe the institution/school needs to reconsider her starting time if she's in need of a break from another job? I'm just wondering really, it does sound strange!

So, is this some kind of music school? Can you tell us more about the actual venue where you are studying?

From the limited information you've provided, I suspect the time has come to talk with owner/manager of music store/school about their policies, and if this is indeed their policy, you must decide whether to continue or to move on.

I often let the students warm up for about 3 to 5 minutes, for several reasons.

First, the piano in my studio is not the piano(s) on which they have been playing all week, so they need a few moments to get re-acclimated to it. Beginners need this even more, as they most likely have not played many pianos, or the piano in general.

I know I need a few moments to adjust when I start to play an unfamiliar piano.

This applies even if I have spent a few hours earlier in the day practicing on a different piano.

Second, students are coming in from what is usually a busy day of other activities and traveling to the studio. It takes a few minutes to center and focus in the studio with the different piano (see above), and different room, etc.

Some students are ready, but others from time to time need those moments to focus and adjust. Pros warm up all the time...Whenever I play with a band, we always have a song or two with which to warm up.

Third, a lot of students have a degree of anxiety playing in front of others, especially their teacher. . . "I played it better at home...honest" is something teachers hear all the time.

So playing for a few minutes alone in the studio gives me an opportunity to hear them without my presence in the room.

No, it is not appropriate for your teacher to short-change you. Maybe she needs a break between lessons. In which case, she should schedule them 15 minutes apart. The psychiatrist's 45 minute "hour" isn't fair in a piano lesson.

This all seems reasonable to me. I would add, though, that part of the lesson could be about what happens at the start of a performance, on an unfamiliar piano, in a place that isn't home. As pianists, we deal with this on a regular basis. When I was an undergrad I complained about this to my teacher, who was herself an experienced concert pianist, and she shrugged, saying "that's just part of the business."

True, and I definitely explain that to students. Good point.

It is a music conservatory that provides studios for instructors, practice rooms for students, and courses (e.g. theory, music history, etc.) that people can sign up for. It is loosely affiliated with a local university, but the instructors and classes are all open to the public.

Even the psychiatrists hour is 50 minutes... Those 45 minute psychiatrists probably spend the first 5 minutes standing in the kitchen and drinking water while they listen to you warm up...

Do you make your check out to the conservatory or to the teacher?

Conservatory.

In that case, I would discuss this matter with the director of the Conservatory. It reflects badly on his or her institution if instructors are just chopping 10 - 15 min off lessons. If, on the other hand, he approves the behavior, I'd start looking for alternate learning opportunities!