We did that. We didn't have any medical issues so don't know if we would have had problems otherwise. When I retired my husband was still not 65 and so didn't have medicare but would turn 65 in a couple of months and he was already retired and on my insurance. We traveled in Europe during that period and so we had travel insurance with medical coverage for that bridge before medicare kicked in.
Would there be any issue with pre-existing conditions by going without for a month if something happened during that month? With medicare we didn't have that issue, but you might if something happened during that month that meant you couldn't go to school. I would investigate how you could get insurance if the worst case scenario happened.
Also look at COBRA. You can sign up for COBRA and not pay for about 60 days. If you end up not needing the insurance you just never pay but if you have an issue, you can pay and COBRA kicks in. We did this so if my husband had had a major issue during that two months, we would have been able to use my old insurance on COBRA until he was on Medicare. Do your own investigation of this, but that is my understanding of it. You can get COBRA from your job and this provides the bridge.