For sure a phone in an aircraft can "see" far more base stations than the system was ever designed to be able to cope with. If it "crashes the network" though then there is something seriously wrong with the network! At worst it would cause interference to some other calls.
The other day I was on Mount Hamilton, just SE of San Jose at 4400' with a fantastic view over the whole south bay and Santa Clara valley. I tried to use my phone. It showed a good signal (no surprise) but every time I tried to make a call, it failed and showed no signal for a few seconds. Similar phenomenon I guess.
Of course 4400' is a lot higher than you normally get to fly small planes in UK airspace...
n5296s