I believed you can. The easiest way is using the network. Just Cut/Copy then paste. Then problem solved. But if you can't connect both computers you can transfer using portable drives. If not you can install your old hard disk to your new one, just make sure that you make the old one as secondary slave and booting from your new one. With this your old hard disk will be considered as a storage device then you can transfer all the files that you need.
Cheers!