I've had a credit card for a long time, and every so often they've sent letters telling me they're increasing my available credit. Does it negatively affect my credit score to have a high limit, one that I never come close to on this card?
No. Higher credit limit is good for your credit score. It's a good sign that the credit card company thinks you are low credit risk. Especially if they are not pulling your credit reports, enjoy the higher credit limit. Keep in mind that in general the higher your total credit line, the lower your debt, hence the lower the "utilization", the better it is for your credit score.
The ratio of debt to credit (balance divided by limit) is more important than the amount of the limit. Missed payments, too many credit accounts, and carrying too high a balance can all negatively affect your score. However, if you own a handful of accounts and keep a low debt-to-credit ratio, you're credit score will be fine.
High credit limits probably won't hurt your credit score, but there can be some repercussions.
When you apply for a home loan, the lender may look at your available credit and total all your credit lines. This may lower the total amount of money they are willing to give you, or make you look like a riskier borrower, increasing your interest rate.