Given an adequately random sample, 1000 is a perfectly valid sample size even for the national electorate, and actually accuracy really doesn't increase if the sample size gets much larger than 2000.
Have you taken, say, an introductory statistics course?
Credible scientific polls have difficulty being correct when people are rapidly changing their minds, but otherwise, despite the abuse they take here from people who would rather believe their own wishful thinking, the general lesson is they are remarkably accurate.
Other than completely worthless straw polls and online polls, the only actual "rigged" poll I've seen that was supposedly legitimate enough to make the RCP listing was the Insider Advantage poll for Newsmax that purported to show Florida "breaking hard" for Newt.
As you say given an adequately random sample, which can be biased in numerous ways in selection.
A poll measuring the stupidity and/or gullibility of the American electorate would be just as accurate as asking them who they will vote for.