The only time I've seen weather interfere with satellite reception was when there was a blizzard, and the winds blew snow that covered the dish. Even then, some channels still worked. The fix was simply to use a broom to gently brush the snow off the dish.
Otherwise, I've had satellite and watched during wind and heavy rainstorms - no problems. I don't get thunderstorms very often, so I don't know if those could cause interference.
Now, satellites ARE susceptible to sun storms. These cause intense bursts of electromagnetic radiation and can interfere with satellite signals (not just TV/Internet). The satellite TV companies used to send out a message whenever one of these storms was detected, to forewarn of any disruptions in service. Fortunately, most of the time the disruption is pretty short - maybe 5-10 minutes.