I bought a bottle cage for my hybrid and I don't know why but I used the plastic screws it came with rather than the metal ones already in my frame. I then knocked the cage and the screws snapped off - leaving me unable to use the frame for a bottle cage. I've taken to two cycle shops but their only suggestion was to have the frame re-threaded but both shops were adamant they were metal screws (they definitely weren't).
Any thoughts on how I could get these out? After getting a kitchen blow-torch for my birthday a thought popped into my head....... but then I imagined black melted paint patches on the frame and thought again.
If those screws were indeed plastic, Pale Rider has solved your problem. However, I suspect they are aluminium. Aluminium screws broken off in the frame's rivnuts (those crimped threaded inserts in the frame) spell trouble and is a job for a skilled mechanic. Aluminium, through galvanic corrosion, swells up and jams in its available space. This prevents it from turning. A plated steel bolt in there can actually be made to turn using a sharp instrument of sorts, but aluminium not.
Several bolt extractors exist, including the Ezee Out type that comprises a tapered carbon steel, reverse-thread device, a Grabbit and fluted rods. All of these require good access, which the frame's inner triangle does not provide. Imagine getting a drill machine in there and then drilling with precision into that small 4mm bolt. It is virtually impossible.
The solution is to remove the rivnut by cutting the flange that lies flat on the frame in three or four places, folding it up like a flower and dropping the rivnut into the frame. This then comes out the bottom bracket. After that, the rivnut has to be replaced.
I would hesitate taking the bike to anyone who hasn't done this before. The fact that they perhaps own a Ezzee Out is not a good indication that they know what they're doing. As Martin said above, breaking an Ezzee Out off inside the bolt is a disaster. They are extremely hard, but brittle and nothing in turn can drill those.
Let's hope it is indeed plastic in there.