YES. It absolutely depends on the fuel and the fuel temperature.
First, avoid "regular" butane and only use isobutane/propane blends. In the US, Snow Peak, MSR, Brunton, and I believe JetBoil should be fine. Don't get Coleman, Optimus, Glowmaster, or Primus for cold weather use.
Second, start with a warm canister. Keep the canister in you bag/quilt at night or in you jacket pocket or something before dinner.
Third, keep the canister warm by keeping the canister in liquid water. Liquid water will always be above 32F/0C. If your fuel vaporizes at 11F/-12C, then you're about 20 degrees Fahrenheit/10 degrees Celcius above the vaporization point of the fuel. You'll have plenty of power. As long as that water stays liquid, you'll have good canister pressure, even if the air temperature drops below the vaporization point of your fuel. Like I say, it's the fuel temperature that matters, not the air temperature.
Adventures in Stoving