I think it's possible because you get the weather that you had in your town while updating. But I haven't ssen a town with heavy snowfall yet, so I'm not sure.
I was curious about this as well since I'm working on a winter town myself, so I set the date to Dec 9th. According to thonky this is the day of the blizzard! I wasn't impressed though
The strongest was on December 10, which was the blizzard day. It's not as exciting as it sounds unfortunateky
Thunderstorms on the other hand are actually really cool since there's lightning and thunder sounds that make it seem like you're in a real storm. Blizzards in new leaf are more like heavier snowfall, more than flurries, but that's about it.