I've switched from left to right to left to right. Of course, the first switch (left to right) was when I was 5 or 6 and broke my left arm and I only bowled once every month or two. I bowled seriously after I turned 7 or so, and stayed right until I was 13 or 14, of which I was around a 180 average. I switched to left for a season and managed to average 165 or so, but have since switched back to right handed.
Being a natural lefty, I've always figured I would (eventually) be a better bowler left-handed than right, but decided against doing so now as I didn't want to miss out on youth tournaments/high school bowling and such when I figured it would take me at least a year or two to get to the level I was/am right handed. I still occasionally bowl left handed when I'm practicing for hours though and find no problem in getting/averaging 170 or so.
With all that, just because you can't support the ball with your opposite hand/arm doesn't mean you're out of the game. I know a few local bowlers that don't even use their arm to help them. One gets the ball set normally (chest/waist level) and pushes out normally and has a free swing. The other starts with the ball straight down/arm straight down, pushs up and out, and has one of the free(est) swings I've ever known. Both average over 200.
It's not impossible to do that, but it does require quite a bit of muscle and concentration/focus on every shot not to have your muscles take over, but it is possible.
In my bag:
Roto-Grip Cell Pearl
Hammer No Mercy Beat'n
Ebonite Total NV
HG - 289 (dang 9 pin)
HS - 746
'09 Summer Avg - 219/178 (adult-jr. house shot / sport shot)