I saw a commercial on the news recently that struck a cord. It was for a phone app that "keeps your family safe in a storm". I then saw an ad for the same app on that local news station's Web site.
I'm a trained weather spotter, a member of the local Weather Amateur Radio Network, and a member of the local ARES/RACES chapter. I know NEVER to rely on a cell phone during a disaster. We had earthquakes in Cincinnati that caused NO DAMAGE and the cell phone network got slammed (because of everyone tweeting/etc about the quake). If that's the case, what do you think the cell network would do in a tornado? We saw it on March 2nd - it went down.
The only safe thing to do is to have a weather radio!
I have a Reecom radio I bought from Amazon. It works well, and I was able to configure it to only do alerts for Clermont County (my county) and only for major warnings - it would blink (but not sound an audible alarm) for flood warnings (which do not affect my home) and severe storm warnings (important, yes, but I usually already know about those).
If you want to go one better...
If you want to go one better, you can get a scanner and listen in on the ham radio ops (we're on 146.88 Mhz and 147.375 Mhz in Cincinnati during severe events). Of course, you're only getting part of the story because police use their own radios to report (you can get scanners for that, too), and weather spotters can also use their telephone to report weather events.
Should you not download the app?
I'm not saying not to download it or use it. If you're in the office or at home and your cell phone is using wi-fi (and you have power and Internet), you're probably fine. I'm saying that once those lights go out, you should already have a weather radio. Don't let one of these phone apps give you a false sense of security.