Woohoo, I can't believe it. My Galaxy Nexus phone is affected by a bug which causes the phone to slow down and to behave very laggy after a while. Some users reported this might happen when they uploaded or downloaded huge files (e.g. movies).
I can't tell whether huge files caused this problem on my Galaxy Nexus, but from one day to another the phone was reacting very slow. It was no longer usable for me. Typing a single character on the keyboard was a pain in the ass ...
I started to google around and found this posting on XDA which saved my phone's life ;-)
If you are experiencing the same behavior on your Galaxy Nexus you might consider following the steps there. But be careful not to HARD BRICK your phone, because your phone must have a special chipset.
My phone was produced in 08/2012 and really has the "suspicious" V3U00M chipset mentioned in the XDA posting. I checked it using the eMMC Brickbug Check app from the play store.
If your phone is not rooted you can't fix the problem. Either you root it now, or you send it to Samsung and let them repair/swap it. It's up to you.
The next thing to do was to run the LagFix (fstrim) free app or paid version. I now use the paid version, because it has a built in scheduler which runs fstrim every night.
The script mentioned in the posting isn't necessary if you use the paid version of LagFix.
After running the LagFix app my phone was reacting as fast as on the day I bought it and it remains fast until now.