I'm a computer programmer by training, but usually when I meet people I tell them I'm a used car salesman, to try to make a better impression.
I'm often ashamed to tell people that I work in the software industry. Computers are just so hard to use.
It is humbling for any software designer to do formal usability testing, to see just how awful software is, even with the best intentions.
A lot of computer programmers like to call themselves "software engineers" (making a big distinction between "programmers" and "engineers") but I still think of engineers as people who get their hands dirty, and who can predict schedules etc.
If you think you might want to be a programmer, some people will tell you to study math and to take certain courses. Nonsense. I studied lots of math (including at the graduate level) and I think 99% of programmers never use any math beyond what they learned in 6th grade. I think music composition and arrangement is much more like programming than is math.
If you want to see if you would like and be a good programmer, buy this book (or better yet, find an old first edition) and read it, doing all the little exercises at the end of each chapter. If you have fun and can do them, you are a programmer. If this is stressful or you can't do them, you might have a hard time becoming a programmer. (Concept from Joel.)