The concept of "bad money driving out the good" was recognized long before Thomas Gresham, but his name is lent to the theory because he persuaded Queen Elizabeth I to restore an already debased currency in 1558.
Simply put, Gresham's Law is the theory holding that if two kinds of money in circulation have the same denominational value but different intrinsic values, the money with higher intrinsic value will be hoarded (for a complete description, see here).
Of course, the action of "hoarding" must first take place, but the United States is close to an environment today where this may return.
For example, copper settled at $1.5126/lb on Friday, February 18. Using the calculations below, the pre-1982 cent that weighs 3.11 grams and a metal composition of 95.0% copper/5.0% zinc has a metal value of $0.01006437 (roughly 100.64% its denomination value).
(.95 × 3.11 × 1.5126 × .0022046) + (.05 × 3.11 × .6186 × .0022046) = 0.01006437
From personal experience, an average roll of circulated cents from any bank will include around 12 to 15 pre-1982 cents (around 25%). Obviously this varies, but I wouldn't consider this a coin that is broadly hoarded...yet.
Keep in mind; the U.S. Mint fabricated two compositions in 1982, the predominately copper version that weighs 3.11 grams and the predominately zinc version still made today that weighs 2.5 grams. The zinc version currently produced isn't quite there yet, but its intrinsic value is beginning to rise with zinc prices hitting new highs. Using Friday's settlement price of $0.6186/lb, the post-1982 cent is worth $0.00353 (roughly 35.33% of its denomination). Zinc prices have a way to go.
The only other interesting trend is nickel. The five-cent piece is composed of 25% nickel/75% copper, with a total weight of 5.00 grams. Using Friday's settlement price of $7.0686/lb for nickel, and copper's $1.5126/lb price, we get the following:
(0.75 × 5 × 1.5126 × .0022046) + (0.25 × 5 × 7.0686 × .0022046) = .03198
A nickel has a current intrinsic value of $0.03198 (roughly 63.97% of its denomination). Nickel and copper prices would need to rise almost 70% before we begin to see a higher intrinsic value than denomination. A long way to go, but certainly plausible.
I guess the big question is, should you go out and trade your paper dollars for common circulating coins (specifically nickels and pennies)? I don't think it's a crazy thing to do in moderation. I wouldn't walk into your bank with a wheel barrow, demanding $10,000 in coinage. I think you should save what you already have though.
Another point to remember is what happened during World War II. The composition of two coins changed during the war; the nickel and the copper cent. Silver (a once plentiful metal) replaced nickel, and steel replaced copper. If you're convinced more war is in the future, converting that $10,000 lying around might not be a bad idea.
For a complete chart of intrinsic coin values, visit coinflation.com