Bicycle Mechanics - Why did my chain fall off?

Bike is an '06 Schwinn S-25 with Shimono gearset. Chain fell off the innermost sprocket at the rear wheel and became quite stuck between sprocket and frame. I was able to pull over and free it using the wheel quick release skewer.

It happened while I was putting power down while shifting from 5th to 7th.

I was really enjoying the transmission on this bike vs. the grip-shift on my previous ride, until this happened.

So I'm left with one of two conclusions: either I can't pedal hard while shifting, or I have to adjust my rear derailleur (after studying French spelling, of course). Which is it?

Two things are usually to blame for the chain coming off the cogs. Ist is an improperly adjusted limit screw on the rear derailleur. Lots of novices like to fool with these for some reason. The other is a bent DR hanger.
Easiest to check for either condition with the chain off; you should easily find derailleur setup instructions on the Park Tool site.

Aha, thank you for the quick reply. So, the derallieur should be adjusted such that even under maximum frame flexure, the chain stays on, eh? Then this will be the fourth error the assemblers of the bike made for me to correct. :)

I went out and located the (well-labelled) L screw. Not going to adjust it now, since it's dark and raining here, but hopefully, in the morning I'll find a setting that lets me access 7th gear, even while pedaling hard.

Two things are usually to blame for the chain coming off the cogs. Ist is an improperly adjusted limit screw on the rear derailleur. Lots of novices like to fool with these for some reason. The other is a bent DR hanger.
Easiest to check for either condition with the chain off; you should easily find derailleur setup instructions on the Park Tool site.

Ditto what he said but if the bike is a new build and you haven't wrecked or dropped it on the right side the derailleur shouldn't be bent.

The limit adjust screws are fairly easy to set. The H limit should be set so that the upper derailleur pulley is centered under the outside edge of the small cog. The L limit should be set so that the upper derailleur pulley is centered with the large cog. After doing that all you need to do is set the b-adjust, located at the top of the derailleur where it mounts to the frame, so that you have about 1/4" seperation between the teeth of the large cog and the upper derailleur pulley. You'll probably find that is set properly. Shift the bike from smallest to largest cog and back and make sure that it shifts cleanly to each cog. Now go for a ride, you're done!

In addition to what the other guys said about the high limit screw, no you cannot pedal hard while shifting. If you here a "crunching" sound while your shifting, you're pedalling too hard and eventually you'll bend sproket and chainwheel teeth sideways.

Just like your car, you have to let off the gas momentarily in order to perform a good shift. Instead of gently moving from one cog to another, excessive pressure will cause the chain to "snap" inward once it clears it's original position. I've had chains spontenously jump off bigger cogs and jump to the lowers while I was downshifting (should go to the larger cogs). This is probably from excessive pressure being applied at the wrong time.