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At times they do! Usually squeezing the blade/scales tightly and drilling lightly will cure it!

I have found a link to better describe what I mean by split-tip bits, if some don't know.
The Difference Between Split-Point Drill Bits & Standard | eHow

I need no center punch or...

Well, I meant as the 1/16 hole is drilled thru the middle, there is a ring of pin still peened holding the collar on. A slightly larger bit will sometimes cut this out and free the collar. A light...

I have used a bit larger than 1/16 to nab the collars, 5/64. I have found it better to wobble with the 1/16.

Yeah, but guys, the center of the pin must be drilled out. Keep that in mind. Don't trash the collars to get a starting point!
Drill thru the middle!

I used a snap center punch for a while. They work good, but they must be unscrewed and 'tuned' to give a lighter 'tap' by grinding off of the main spring. They still want to go where you don't want...

Nice work! These things which have been coming up are so useful to do accurate restos. Thanks to all of you which contribute!
Relieves me to see someone else coming up to the bit. Works for...