Slacker,
Maybe you are OK, maybe you are not. Apart from the type of derailleur (i.e long or short cage), it depends mostly on the age of your current derailleur and shifters. If they were made for 5 or 6 gears, it's likely they won't work with 8 to 10, but if they were made for 7 gears, it's likely one or both items will work.
Things to watch for:
1. Derailleur
You might be able to use it by unscrewing both limit screws. But if it was made for a freewheel, check inner clearance. Old derailleurs required a bit more inside clearance, and it might hit the spokes when you are in low gear. If it is too close for comfort, change it.
Also, pre-1985 derailleurs were less precise than modern ones, so you might be better with one of the cheaper Shimano derailleurs than with the one you have.
And by the way, a new derailleur will work in indexed mode but the old one wont. So if you change the shifters, make sure you either get new shifters that work in friction mode (i.e. forget STI) or change the derailleur too.
One way to check derailleur before you change the cogs is to remove the chain, unscrew both limit screws and see how far the derailleur moves.
2. Shifters
If the shifters bottom out with a 7-gear setup, they won't work with 8, 9 or 10 gears. But if they are like mine, on a 7-speed setup, the shifters cover a 150-degree arc whereas they could cover a 180-degree arc (i.e. if I remove the cable, they do a full half-circle). Therefore I assume my shifters could work with 8 to 10 gears.