This might sound kind of odd... but it drove me to the brink of insanty for years.
There is/was a bug in WinUAE with the bsdsocket.library which made WinUAE behave very strangely. I've not encountered it as of lately, but me and many many others that I've talked to have experienced this very bug, even in versions of WinUAE where it clearly stated that the bsdsocket.library was fixed.
Well, anyways, on to the bug itself... what happened was that when trying to access internet or anything related to TCP/IP it didn't seem to work. Though, if hitting the config button (to config WinUAE) and the ESC, hitting it again and then pressing ESC it did work. So, to download a tiny little file could involve pressing F12 and ESC like 20-30 times each. Very annoying.
But, like I said... I think this is fixed as of now. And besides me and my friends having seen this, I have never heard anyone else talk about this very bug.