Or can one make that buffer very big - lets say 16 kbytes?
No, that's impossible because your processor only has 2kB of RAM.
I guess it would be nice if in the documentation it was mentioned that one gets buffer overruns that easily... and that this can be remedied (in some cases) by making the buffer bigger.
That's not a problem of the buffer, it's an inherent problem of SoftwareSerial. It should be documented to not use that class unless you exactly know what you're doing and you have no other way to solve the problem of having a missing UART.
The SoftwareSerial class is blocking interrupts during the complete reception or send process. For the hardware serial interface interrupts are used to feed the next byte from the sending buffer into the hardware register or store the next byte from the hardware register in the reception buffer. So it's kind of a dilemma not easily solvable. If it's possible in your case try using AltSoftSerial (AltSoftSerial Library, for an extra serial port) because that implementation doesn't block interrupts but uses a timer and is limited to fixed pins.