Yes with any indexed shifter.....
you should be able to run through the gears just working the lever both up and down. This is not an uncommon problem with shimano shifters. The factory lube becomes quite with a little dirt and dust added, and disuse, becomes quite sticky over time. What we do at the shop when we run into this problem is pretty simple. Remove all of the covers that you can without actually disassembling the shifter mechanism, you'd never get it back together right if you did. Now you have two choices, either douse the mechanisim with degreaser and use an old tooth brush and scrub it out, carefully. Then if you have compressed air blow it dry then relube everything heavily with something like Triflow or other light oil. Or just douse it with Triflow (and I mean get it dripping). From there work the shifter until it frees up. What happens is, there are little spring loaded pawls that engage ratched cogs inside the shifter. When the lever is moved in one direction, usually the down shift, the pawl engages the ratchet and moves the cable by moving the cable seat. When moved the other way a pawl moves a lever that disengages the down shift pawl from the ratched cog and allows it to move one nothch before spring tension snaps the pawl back into place. This releases tension on the cable. This is highly over simplified as there are a couple of other items involved as well, but it's the general idea. So when the factory lube gets sticky and stiff, it prevents these pawls from engaging as they should. The springs that provide the tension on the pawls and other bits are quite small and not overly stiff. So when the lube gets sticky it prevents the springs from doing their job and the pawls can't move as they should. That's why you're not hearing any clicks when you shift, things aren't engaging properly. I know we've done several dozen bikes this season at the shop with the same problem, more than in the past. Likely because of the high gas prices. People are diggin out the 10 year old clunker that hasn't been ridden in 2 or 5 years and wanting to get it in shape to ride. So we've seen allot of them.
Anyway, a good cleaning/degreasing and a relube should solve your problem. Just one trick that I've learned that helps. Get a pice of old shift cable and cut it to about 4 feet long. Once you've cleaned and lubed the shifter, mount it back on the bar without the covers installed. Insert the cable in the seat as you would a new cable. Then wrap the cable around your hand once and run through the gears putting mild tension on the cable like your derailleur would. You'll be able to feel the shifter working and you'll be able to feel any sticky spots and you'll be able to feel any hesitation in engagement. If it feels crisp and positive though the range both up and down, you've done it right.
Good Dirt