I am unclear, but I think the God doesn't give you more than you can handle thing is a misquote. I Corinithians 10:13 talks about not being tempted more than you can bear, but that's a whole 'nother kettle of fish than suffering.
Not exactly, the original Greek carries the meaning of suffering a trial (the root verb means "to pierce through"). I think you're right that it gets misread these days only to include suffering things that we regard negatively, by which I mean it doesn't get applied as naturally to being, for want of a better word, bribed into wrongdoing. (And I'm sure Paul regarded resisting such temptations as worthy of being deemed a trial.) However, I don't think it would be accurate either to read it as not talking about the trials of faith that arise from suffering (such as despair, rage, self-pity, what have you).
Now, I do think it gets misused. It was clearly intended by Paul to encourage people facing trials (whether of the ouch ouch make it stop variety, or the sexy funtimes this will go straight to your hips variety). I doubt he would've approved of people then judging how well others managed their grief/pain/rock'n'roller lifestyle. ("I give it an 8. Decent technique, a bit of flair, but shaky on the dismount.") The verse is relevant to the notion of divine judgment, however, insofar as such judgment becomes unjust if the subject were in fact unable to escape becoming a sinner. Given the tendency of people to judge t IRONY ALERT , it's not surprising this verse gets misused to look down on people who have been broken by suffering.
Another direction in which it gets poorly applied is by conflating it with a prosperity gospel, and here we really get to blame the victim. Because then, if you suffer some loss, and you don't actually bounce back but your life turns to crap thereafter, must be because you didn't maintain a positive attitude and treat it like the growth experience it was obviously meant to be.