Using the usual pin vices I used to break drill bits all the time. I switched to the carbide bits that come with a shank and rubber collar to hold and spin bit, but still I broke them.
No matter how careful, I was introducing a shear pressure as I rotated the bit.
Then about 5 years ago, I bought a fairly inexpensive USB mini drill off ebay. I think it was about 12GBP but when I then bought one for my son, a year or two later, price had gone up to 18GBP or so. I haven't checked recently, but I'd pay twice as much if necessary.
The drill has variable speed settings and is light and easy to hold - like a paintbrush. One USB charge lasts me weeks.
To use it, I make a slight impression in the work with a steel pointy tool, then switch on drill at lowest speed and gently apply to the pre-marked spot. Since I am only holding the drill I am applying no sideways pressure whatsoever so no shear stress being applied to the bit. In it goes like a hot knife into butter and out again. There can be a little melted plastic on the bit afterwards and I carefully slice this off with a scalpel while bit is held flat on cutting mat (careful not to break bit here!)
Don't try to drill without the starter mark or drill will skid all over the place.
I don't think I've broken an HSS or carbide bit so far, on any work using this magic wee drill.
That's my tip for saving drill bits FWIW!
Sandy