No doubt that flat screen TV gives you a better resolution but is there truth to the rumor that the possibility of flat screen TVs having problems or breaking down would be higher as compare to regular screen TV?

I have two flat screen TVs - a Toshiba 20" and a Sony 34" widescreen.

This is false. Flat screen TV's do not give better resolution or a better picture. What they do is help reduce glare and provide a wider viewing angle. They also look "cool" compared to curved screens.

As far as I know flat screen TVs do not break or have any more problems than curved tube TVs.

If you are talking about flat panel (i.e. LCD,plasma) they are prone or at least at risk of having dead pixels...

And if you're talking about flat screen CRTs, they have a harder time focusing at the edges than curved screens and tend to have convergence problems there too. You'll often see other geometry aberrations at the sides and corners as well.

But the other part of the screen can provide a very nice picture!

While flat screen SD crt's have the same 480i resolution limitations as curved screens....my perception is that flat screen tend to "look" sharper than rounded crts. My 36" Flat screen Toshiba looks sharper than my old 32" SD Sony XBR ever did.

A lot of the picture quality has to do with the rest of the TV as well. Most major brand flat screens do provide cleaner signal processing than a plain round tube. As noted, the screens (flat or round) have pretty much the same display properties.