Cheers for the answers guys.
I will try putting some delay between the initialization of the servos and reduce the moving speed to see what happens.
Nothing is connected to pin 1 now, the code is slightly outdated (it was up to date when I posted it).
I have tried simple routines with the servos (with one and six at a time) and never encountered such a problem. This only started happening with all 12 running at the same time. And on the first attempts it wasn't as bad but lots of things had to be mechanically adjusted and eventually it got worse.
I will see the robot again on Tuesday afternoon, so that's when I'll post an update.