- Is it normal for new graduate students to feel very overwhelmed and discouraged when they first start their program?
Yes, absolutely. 1000000x yes.
- What happens when a graduate program realizes they've made a mistake in admission? What would qualify as a "mistake"?
A "mistake" is admitting a student who is not capable of meeting minimum standards in the department (or at least, not now), and is
therefore dismissed from the program
for academic reasons. This is an unfortunate mistake for both the student and the department (as it represents quite a bit of wasted time, money, effort, and emotional energy on both sides), and so we try very hard to help students who are in the "danger zone" to recover and avoid dismissal.
Generally, a graduate program has very specific regulations describing how and under what circumstances a student is dismissed from the program for academics. In my program, this starts with a semester of academic probation, so students have some warning that they're in danger of dismissal and have a chance to bring their grades up. Many students do manage to bring their grades up after a probation semester and successfully complete the program.
- If I'm feeling discouraged and/or overwhelmed by my courses, would it reflect poorly on my reputation in the program to speak to my advisor about it?
In my department, it would reflect well on you. It would show a maturity, self-awareness, and sense of responsiblity that your advisor probably wishes every student in the program had.