While the advice on annealing non ferrous metals is certainly good to know, hardness ain't your problem.
The bits are grabbing and breaking because of copper's "sticky" or gummy quality, as mentioned above. Silver is the same way. To prevent this you need lubrication. (actually, not so much lubrication as a barrier between the metals.) In my fine metals classes, we were taught to keep some wax (Surely your wife has plenty of scented candles- she'll never miss one!) handy and just rub it on the spinning drill bit before lowering it to the metal. You gotta keep backing the bit out and reapplying it to the tip & sides of the bit. Since this way is less messy than oil, it's all I've used, but if you're more accustomed to oil or cutting fluid, I suppose that should work just as well. It's always a good idea to keep the chips cleared, but as long as you lubricate they should not be a problem. I've never had any reason to grind the tips off my bits this way.