From what I understand, the Nexus 4 doesn't support USB Mass Storage mode when connecting to your computer via USB (which would let your computer see your phone storage as another drive)--only MTP and PTP. MTP is used for moving music files, and PTP for picture files--see this
article explaining MTP. First of all, make sure that your USB connection is set to MTP, not PTP. You should be able to tell by swiping down the notification shade after plugging in via USB, and selecting the USB connection notification. If it's set to PTP, then select MTP.
If it still doesn't work, I think the problem with MTP (Media Transfer Protocol) may be that it blocks transfer of some files if they're felt to be incompatible with the native player, since MTP was originally designed as a way to sync music between a computer and an MP3 player. To get around this, consider an app like
AirDroid, which allows you to move files wirelessly, without any USB connection.