Homebrew PCBs

I just got a set of resharpened carbide bits in from drill city. I
chuck one up into my dremel press. I snap the bit on the first hole.
Thinking maybe I moved the board, I tried a second bit and snapped it
as well. This was with a #81 bit. So I thought maybe I am just a
little clumsy and try a 1/32 bit and snap it. Noticeing that the bit
did not penetrate all the way, I thought maybe I needed a backer.
Still no success and I snap another 1/32 bit.

My guess is that the dremel just has too much slop for these bits. Any
suggestions.