Currently split from ex. Neither claimed our child last year. Is it possible to go back this year and claim her ? It would be 2016 tax year

it depends; your ex claimed the child, but If there is a court order and say it says yopu are supposed to claim the kid then, you can do that but remember: There are several things you need to be aware of: The IRS doesn't care about your court order, unless it is dated before 2009. The IRS goes by it's own rules and will award the dependency to the custodial parent, if both parents try to claim the child. Your only remedy against the ex is to take her/him back to court for sanctions; You would then need to file your mended return on paper, claiming the child as appropriate. The IRS will process your return and send you your refund, in the normal time. Shortly (up to a year) thereafter, you'll receive a letter from the IRS, stating that your child was claimed on another return. It will tell you that if you made a mistake to file an amended return and if you didn't make a mistake to do nothing. The other party will get the same letter you did. If one of you doesn't file an amended return, unclaiming the child, the next letter, from the IRS, will require you to provide proof. Be sure to reply in a timely manner.

if you can claim your kid on your 2016 return then you must file amended 2016 return; to claim you r kid on your past 2016 return, Your child must live with you for more than half the year, but several exceptions apply. And youmust provide more than half of the kid?s support.
The statutes of limitations not only limits the IRS in assessing additional tax on returns filed, but it also limits the amount of time you have to claim a refund or credit due. If the three-year deadline for filing has passed, the IRS, by law, cannot issue your refund

Our court order says every other year. She did not claim the child last year and would have no problem with me claiming the child for that year 2016
I'm guessing they will only look into the court order as far as who's year or just give to custodial parent as you said if there is a problem where both people are trying to claim the child but if she didn't, isn't planning on it, and doesn't mind me doing so. Can I ? Should I ?

In general, the custodial parent claims the child for tax purposes. If the court does not make any orders about the tax deduction, then the custodial parent automatically claims the child as a dependent for tax purposes. The IRS income tax rules say that the parent having custody for the greater portion of the calendar year receives the deduction.
A court, can, however, order that the non-custodial parent be allowed to claim the child for tax purposes. The court will order the custodial parent to sign the proper IRS forms to allow the non-custodial parent to claim the child for tax purposes. The custodial parent , and provides it to the noncustodial parent who attaches it to his or her return. Often, the court will allow the non-custodial parent to claim the child for tax purposes only if that parent is current on his or her child support payments.