No, you cannot claim child care expenses paid in 2015 on your 2016 tax return.
If you wish to claim them, you'll need to amend your 2015 tax return.
Please read this TurboTax FAQ on how to amend a tax return:
https://ttlc.intuit.com/replies/3288565
**Say "Thanks" by clicking the thumb icon in a post
**Mark the post that answers your question by clicking on "Mark as Best Answer"