you might end up earning less after tax as a manager than you did as an engineer as your salary increase puts you in a new tax bracket.

This is impossible, even with the very high German taxes :-)

You will keep less of each (EDIT: additional) Euro you earn, but the tax progression is not that broken that getting paid more results in less money for you. There are diminishing returns on hours worked, if that is what you meant, but otherwise it's untrue.