Sony support has only offered to me to send my videos on a disc to them to see if they can find out what is wrong... so I decided to get more concrete answers.
I own a Sony digital camera, the Cybershot DSC-HX100V. It can do 1920x1080 video at 60 frames per second progressive (60p) at 28Mbps data (or bit) rate. My Sony home blu-ray player is a model from 2010 - the BDP-S370. It recently got a firmware upgrade by Sony claiming that this new firmware ENABLES the playing of 50p/60p videos produced by Sony cameras. My firmware upgrade went perfectly without issues and showed successful on the blu-ray. My videos still are choppy and basically, stop, play, stop, play with about 1 second stops and resuming of play on its own, so they are unplayable if I copy them onto a blu-ray disc OR if I insert a USB memory stick into the blu-ray player's USB slot. The USB stick is high-speed - the Rally 2 by OCZ - which can write at 15MBps and read at 25MBps.
By the way, if I reduce the quality of the videos on the camera menu from PS (the highest mentioned above) to FX (1920x1080 at 24Mbps data rate and 60p/60i though I think this is 60i/30p because that's how my software interprets it), they play fine on the blu-ray player (BDP-S370). Also, all the videos play fine on my computer blu-ray burner (an LG) and from the hard drive or from the same OCZ memory stick.
I also have a Sony HD TV by the way, the Bravia 40EX400, 60Hz, 40 inch, LCD.
Has anyone experienced this issue? Any suggestions?