Nick, Thanks for trying to help!
I tried lower transmission speeds between the two Mega's down to SPI.setClockDivider(SPI_CLOCK_DIV128); the result was the same. Then I tried faster transmission speeds, the fasted that worked (with the same problems) was SPI.setClockDivider(SPI_CLOCK_DIV4).
Then I tried to transmit 75 characters. That worked a few times with similar problems and then the receiver simply stopped working.
Have you heard if anyone has similar SPI problems for Arduino to Arduino communications (I could not find other examples)? Given the pedigree of the original programs, could it perhaps be possible that there are problems with the Mega SPI library?