Hi Jan,
I have never used the ADXL345 but I am also interested in it.
I am not the GURU but without a printscreen of the scope it is very difficult.
The problem you describe is very strange. I cannot exclude the ADXL345 is broken but, in my opinion, you have to be sure about the environment around.
I think the first things to do are:
1) add some bypass capacitors on VCC;
2) too short or too long tracks could generate glitch on bus signals (sometime hard to capture with oscilloscope), so I suggest to add a resistor or/and capacitor on I2C (also CS).
Thanks for your attention,
best regards.
Fulvio