Can I make a trade and claim a FA at the same time?

For leagues using the Instant Trade option the answer is Yes. You can make trades and free agent claims on the same day.

For leagues using the Trade Review option the answer is No. You have to wait 48-hours until the trade is processed. Once the trade is processed and you have your new players you can make a free agent claim.

This is because in leagues using Trade Review, the game computer will only process one transaction (FA pickup or trade) per team per night you cannot claim a FA and make a trade on the same day. When the game engine runs overnight, the last transaction you made will be the one processed.