You will have to go through U.S. Customs and Immigration at JFK, and claim your checked baggage upon arrival at JFK. If you are not a U.S. citizen, make sure that you satisfy immigration requirements for transiting the U.S. If your carrier into JFK has an interline baggage agreement with your carrier out of JFK, then you can ask the first carrier to check the baggage "through" to SJO; then, you should be able to drop the baggage at a re-check desk right after you clear Customs. If the baggage is not checked through to SJO, you will have to carry it to Terminal 5 after you clear Customs, and re-check it there.
You will have to go through