Yes, using no-name power supplies can damage your equipment.
On the side note, they aren't that bad if you've got a basic set up with 1 HDD, 1 CD-RW, non-power hungry video card and no lights/case fans. Just enough to get by, i.e. CPU fan and PSU fan.
All those mom and pop stores don't sell Antec type PSUs to most of their customers. They buy the $30 dollar cases that come with no-name PSUs. I'm sure they would get a heck lot more returns if they were that bad. But for a basic set up, they should work fine.
Yes, no-name PSUs probably do fail more often than name-brand ones, but I don't think it's as high as 1 out of every 2 or 3. I think it's much less than that.
But it should be ok, as long as you don't have too many things running from it. If you can and need more than the basic setup, get a name-brand PSU. It's safer and more reliable.