Kostya,
I didn't figure out the systematics yet, but I've observed the following behavior:
When working exclusively with the mobile network (no Wi-Fi available), Aquamail downloaded incomplete messages. Then, when I am trying to download the complete message Aquamail says it cannot.
The message itself is rather long (~0.2 - 0.6 MB), text only.
Any suggestions of what I can try to help with the diagnostics?
(although I am not sure if I will be able to reproduce this soon, as I won't be without Wi-Fi for long periods of time again.)