When taking a 5 minute shower, the water in the tub rises to about 2 inches. I've tried chemical drain openers, and also one of those plastic, nubby strips that is designed to open drains. Not much luck with either.
From the illustrations I've seen online, I guess the drain pipe makes a 90 degree turn after going down an inch or so, and in mine I've found one or two small openings on the side. I can barely get the plastic strip past those, and when I do it doesn't bring up a clog. 
I've had better luck attaching the strip to a straightened wire coat hanger, and plunging them both through the overflow opening after removing the drain handle plate. But again, I don't get a clog out.
Should I invest in a real plumber's snake, or just give up and call a plumber? 
Or could it be that the drain pipe is partially frozen (here in Vermont) so we get some drainage but not enough? We don't have this problem with the sinks.
Thanks.

When you removed the 'drain plate' did it have all the pieces shown in this picture? [images.meredith.com]
If it didn't, part of it has fallen into the overflow assembly and needs to be fished out. If it all came out you can attempt to use a strategically bent coat hanger to pull out the hair clog, or simply call a plumber. you may be able to unscrew the strainer on the bottom of the tub and see hair caught up on the crossbars in the drain and pull hair out that way as well. Let me know about the tub drain assembly first and we can go from there.