That's an interesting catch -- the 1uf +5V bypass cap. Indeed, as is my practice, I am using a .1uf there.
The +5 is a 7805 as drawn... Good power supply -- and I have looked at it on the scope. Its clean.
I will beef up that pin 16 Cap with a 1uf and see if I have the same problem.
I looked back at my other boards that use the MAX232 and they all have a big cap on the power supply somewhere. Not necessarily right at the VCC but close by. Could that be the problem? I don't know, but something weird is happening.
Greg