Last time I played this game was probably in somethign like 1987 so I don't remember a whole lot of it. But basically here's what I think you did:

You climbed ladders or something on the outside of a building maybe.
Possibly 8 "levels" (by that I mean that there are 8 horizontal lines that the bad guys are on connected by ladders).
Your dude doesn't walk on the ground, he hangs from the next level up.
I used a joystick, don't know if you can use the keyboard.
The bad guys were just below you, your hanging feet would hit them and you'd die or something else bad would happen, but if you hit the button on the joystick you could raise your feet up for just a bit, long enough to go over the top of them, or you could stop on top of them and squish them. There may have been some dudes you couldnt' squish that you had to skim over.
You'd get to the top of the building and that was the end of the level, I'm not sure if you had to fight anyone or not at the top. But the next level was much of the same, just more bad dudes.

I realize this is pretty sketchy information, but if you have played it before you will recognise it from that I think. If you don't recognise this then you probably didn't play it.

Edit: Somebody suggested that it might be Lode Runner. That unfortunately isn't the game I'm looking for.

Nope, unfortunately its not it. I've gotten a lot of suggestions on our schools message boards for games that look like Jumpman. The thing is though is that the flat areas are continous across the screen, they don't have places you can fall, the only thing that makes one level (by that I don't mean like level 1 level 2, I mean platforms) different from another is the placement of the spots you can climb up.
come to think about it they may not be ladders, they may just be poles you climb up. Not more than 1 inch tall on a 13" screen.

No, I remember that one, it's not that. Montezuma's Revenge is cool though

This is about all I can remember about this, maybe I was wrong about the ladders, but there is a way to climb to the next higher platform.
Notice the platforms are not broken like the 2 games svtcobra suggested. I'm about to give up looking anyway, I've been to a few C64 sites and gone through over 20,000 screenshots and haven't found it. I've probably seen the game in those but it probably was the title screen which if I knew what that looked like I'd remember what the game name was.

Found it!

Squish'em
Here is a youtube link on Colecovision: http://www.youtube.com/watch?v=FtiM5adoS5A

Screenshot from C64:


I kind of remember the graphics being better than that.....


Edit: Found a C64 video of it that is identical to what I remember: http://www.youtube.com/watch?v=0F7wxqhlHLg