The most common problem with veneered Chest of drawers is a blown top.

This is where shrinkage of the underlying carcass wood has occurred and the veneer has lifted and often split. Re-veneering is costly.

Frame shrinkage will result in drawers not fitting properly or doors not closing.

Check all the handles; do they match? If one is missing, it may be difficult to find an exact copy and you will then have to replace them all.  There are companies that make excellent reproduction handles, such as .

Look at the drawer runners.  There may be excessive wear, especially with softwood runners and linings.  These will need to be repaired with new wood.

Pull out a drawer and look at its base. Plywood was used to make drawer bases from about 1920. Inspection of the dovetail joints will help you to date the piece.