If I have crowns on my back teeth, could I still get braces?

You can have braces if you have dental crowns on your back teeth.  The brackets will bond to the crowns instead of your enamel.  This is very common in orthodontics.  

You can get braces with crowned teeth. You need to take care of them like avoiding certain foods to reduce risk of damage to braces. Best of luck

Yes, most certainly!Your orthodontist will just use brackets to hold the orthodontic wire that is glued onto the crown, rather than single brackets that can be glued onto the enamel.  However, sometimes a crown will loosen up, especially if there isn't enough good tooth structure under it.  If that happens, the orthodontist will just need to recement the crown.