For weather to seriously affect a train it would have to be pretty bad weather. I'm from northeastern Montana, where the weather can get really severe. There are times when the airlines can't fly, the roads are closed to all traffic, but Amtrak is still running. It might be delayed a bit for what ever reason, but it's still going strong.
Another example: on one trip from Portland to my parents' home, the Columbia River gorge was an absolute mess. The Interstate through the gorge was completely closed due to some really nasty ice. It was bad. On the other side of the gorge, which was also experiencing the same icing conditions, my train flew along at normal speed like it was just another normal, nice weather day.