Is it a problem that I have two of each color wires on my Honeywell thermostat?
The answer is yes; it was a problem. The wiring wasn't done correctly. I had an HVAC guy come out. He suspected that whoever installed my thermostat cheated by using some lower grade wire and doubling them up. Sure enough when he untwisted the wires from the old thermostat and checked the voltage with a multimeter we saw that the pair would give us the necessary 24 volts but an individual wire would only give us about 17. Nice.

Anyway, I shelled out about $100 to have him run a real HVAC cable. (I had him use 10 wire just incase I want to make use of more of the new thermostat's features in the future).