I have the same problem on my 2013 X5 35i Xdrive glove compartment. I started replacing all the interior lights with LED's and the glove box light was the last one when it happen! I did some digging and found out about a fuse that you can replace it yourself for under $6.00 (5 pack)! The fuse is 10A "minis. This is the site,

Help with Glovebox issue