We are using Moodle 2.5.1+ (Build: 20130815) in one of our sites and Moodle 2.5.2+ (Build: 20130927) on three other sites.
In Moodle 2.5.1+ (Build: 20130815) and in one of the Moodle 2.5.2+ (Build: 20130927) instances, this problem appears... on the other two instances of Moodle 2.5.2+ (Build: 20130927) it does not appear.
This is strange... speccially because it started two days ago only.
Upgraded Moodle 2.5.1+ (Build: 20130815) to Moodle 2.5.2+ (Build: 20130927) and the problem remains.
More strange is the fact that I deleted and reinstalled the Essential theme in the Moodle 2.5.2+ (Build: 20130927) that was not working and it started working. Tried the same in the Moodle that was upgraded and the problem remains...
Other situation is the fact that although reinstaling the Essential theme fixed the problem in one of the Moodles - the Moodle 2.5.2+ (Build: 20130927), another problem remains: the menu bar is acting strange... I have to click severall times until a menu opens. All other Moodle 2.5.2+ (Build: 20130927), not the one that I upgraded, also have this menu bars and they are working just fine.
On the three original Moodle 2.5.2+ (Build: 20130927) I had the Essential Theme... based on Bootstrap, that appears to still have some problems. But two were working perfectly and the third was not.
On the other Moodle (the one upgraded) we use Form Factor, a standard Moodle theme.
Nevertheless, it must be a Theme related subject, because in all Moodles, if I change to the "Clean" theme, it all starts working just fine... menus, filepicker, and so on.
Have you tried changing your themes?