I’m not sure all of the Shimano rapid fires are the same because I haven’t worked on all of the different ones LX, XT, XTR etc. but my guess is that they are very similar. Inside the shifter there is a pawl and gear mechanism. When you push on the big paddle, the pawl rotates and contacts a gear, the gear is essentially attached to the cable and it pulls it. There is a little more to it than that, but these are the basics. The factory goops up the insides pretty good with grease. When it gets cold outside or the inside of the shifter just gets too dirty over time, the condition you are describing can occur. When this happens, the pawl has a hard time engaging the gear. They just need taken apart and cleaned. Just be careful, there are some small parts that may want to fall and hide somewhere on your floor.
From your description, this sounds like this may be your problem.