I have certainly had this experience. It probably happens somewhere between 1/3 to 1/2 of the time. That said, it does not seem to be particular to PS4 games. I've had the same issues with PS3, PS2, Xbox (original), and bluray movies. Sometimes its just loose, but sometimes the spool is broken (which is definitely the much more frustrating scenario).
I will say that I've often (but not always) found that those cases that let go of their discs in transit tend to hold the disc less securely going forward. My guess has always been that there's an almost imperceptible difference in the measurement of the spool's cut that causes the disc to sit ever so much more loosely.
As to the damage, I've never had any issues with a PS4 or PS3 game and rarely with a PS2 or Xbox game. However, I have had several bluray movies suffer scratches due to this phenomenon (which shocked me, honestly, given the density of the discs plastic). Like abronn627, I also used to work for a video game retailer, and, in my experience, scratching BDs was a much more difficult proposition than scratching DVDs. Through this job, I actually found that much of what looks like a scratch on BDs can actually be wiped away with a microfiber cloth - in case that information is ever useful to anyone :D
Currently Playing (So the world might be mended):
|PlayStation 4:||PlayStation 3:||PlayStation Portable:||PlayStation Vita:|
|Resident Evil: Revelations 2 (Raid Mode)||Tokyo Jungle||Valkyria Chronicles III||
Soul Sacrifice: Delta
|Transistor (Platinum run)||Doom 3: BFG Edition||PC:||
|Salt & Sanctuary||Metal Gear Solid: Peace Walker HD||Death's Gambit|
|Deadly Premonition: DC||Deus Ex: The Fall|