I agree somewhat that you shouldn't rely on an internet stream as your sole primary choice. Everyone needs a All Hazards Radio so they can't get alerted and the early warning of emergencies. With that said that, the internet stream can come in handy because if you are somewhere where you don't have a radio like work, home, etc. You can still monitor the weather if you know it is suppose to be or get bad. Worse case scenario is if the power went out, but not everything weather related is going to knock out power. I know here in Texas I get most of my listeners during Winter Weather like ice or snow is coming. Yea I know, sounds strange when we are in Tornado alley, but that just goes to show how comfortable people are with severe weather. So it can come it handy and it is something which is better then nothing.
My Wx PC is dedicated just for weather and runs three HD LCD monitors so I can monitor a lot at one time should things get hairy. Everything is running on a APC UPS should the power go out and the radio for the stream is AC powered with battery back up in the radio. I have two seperate radios in the office, one for the stream and one for my personal use. I agree with Mark though, even if you do have a weather radio it is possible in worse case scenario that the radio feed could go out and TV too like in a Hurricane or what not like has happened in Houston. So it would be a good idea to have a scanner with HAM freq programmed in.