Does the weather have a significant effect on the distance a radio signal will travel?

For examply, if it's extremely humid, will my wireless network not make it as far? If it's snowing, will a weak radio station be harder to pick up?

Yes, the weather will have an effect on radio signals. Some signals will reflect (partially or totally) or change phase when exposed to rain or snow, or even changes in air density -- detecting this is how weather radar works.

Depending on the weakness of an AM signal in a snowstorm, you could even "hear" the snow falling in the signal. It'd sound like sporadic static. A heavy snow/blizzard could cause a complete blackout of signal. I worked at my station during a snow storm last winter, I had to go up on the roof with spikes on my shoes and clear the satellites and the microwave horn, and had to knock snow off of the top of my STL Tower (55 feet above the busiest street in town)