What happens if I overpay a credit card bill?

Yes, credit card overpayments will typically appear as a statement credit in the form of a negative balance on your bill. Making further purchases will bring your balance up to zero, from which point you still have your regular credit limit to work with. Your credit limit of $2000 implies the credit card company is not comfortable loaning you more than that amount - but if you give them $1k before borrowing $3k, you still haven't actually borrowed more than $2k.

There are also ways to request the overpayment as a refund directly from the credit card company, but simply zeroing the balance is usually the simplest approach for a modest overpayment on a card you use regularly.

As pointed out in the comments, it may not be possible to do this in all cases, as some credit cards won't let you overpay. Doing so could be seen as a form of "credit cycling" in which a credit balance is paid down in the middle of a statement period, and then the available balance re-used - it can be seen as a way to artificially spend more than your credit limit within one period, going against the risk profile the credit company has set for you.

tl;dr: It depends on your bank. Call them and ask.

Some examples of different policies include (these are not all the same bank):

1. I had a card I didn't use very often, and I forgot it was on auto-pay. I used it for a decent-sized purchase, and paid the bill in full the day before the due date, but it didn't post until the next day and the automatic minimum payment still came out. I had a credit of about $30 for 2 months and then after I didn't use the card again they mailed me a check. (I believe the policy was a credit couldn't sit for more than 60 days.)
2. I had a business card and I wanted to purchase something that was about 10% more than my limit. I called customer service and was told I could over pay by up to 50% of my limit and then make the purchase, which I did.
3. With that same business card, I wanted a higher limit, so I originally started paying it off twice a month so I could effectively double my limit. I think I did this about 3 times over 6 months and then I requested a limit increase which was granted.
4. I have had cards that would not let me overpay even if I tried, using their online payment system. (I don't know what would have happened if I mailed a check.)

I'm fairly certain that if you end up with a credit (either from overpaying or from returns/disputes/etc), then the credit will effectively raise your limit the next month, and, if you let the credit sit long enough they will refund the money to you. Again, your best course of action is to call the number on the back of your CC and ask, as bank policy differs greatly on this.

Your question is focused on the situation where the bill you are trying to pay is equal to the maximum available credit.

A more common situation is when somebody pays more than the amount owed, but they were not near the credit limit. Many people use a credit card multiple times per week, which means that by the time they pay the bill just before the due date there are two or three weeks of items that would be due on the following bill.

Most of my credit cards have a standard option to pay what was on the bill, or the current balance (which is a larger amount). If you are in this situation the amount you owe on the next bill could be reduced, or completely covered.

There are other situations where you pay more than you owe, because you returned an item between when the bill closed and the due date. I generally just pay the larger amount because the excess will be applied to the next billing cycle.

Some people pay their bill early, so that they can create space on the account. This make it easier to handle an emergency. Sometimes they pay what they owe before the bill is even generated. Operating this way on a regular basis seems confusing, but I can understand it on an occasional basis.

Some people overpay to be able to charge an item more expensive than their credit limit. Sometimes it works, sometimes it doesn't. Eventually if you never use the over payment, they will refund the money.