I have been fortunate enough to be good at spelling without having to work at it. To unxpurg8d's point, I read a LOT. My wife used to be an atrocious speller, but since she has started to read much more, her spelling is greatly improved.
I think there is indeed a connection there. I don't spell words by memorizing the spelling, I can look at the word and "see" if it's right or wrong. That must come from reading. The words I tend to misspell (words ending in -ance/-ence among others) don't "look" wrong to me when I screw them up, and the spell-checker saves me in those cases.
Spell checkers are no substitute for knowing which word to use, though, since they can't save you when you use a correctly-spelled wrong word. Our local paper runs a column about spelling/grammar questions that features mistakes people spot and send in called "Typo of the Weak".
Some examples:
"We proudly feature some-day shipping."
"Our massage treatments help relive your pain."
"Sign up now for our Beauty and Fitness Curse."
"The penis mightier than the sword."