All games are different so saying "this plays fine" is no a good guide. Destiny needs a high quality internet signal and in fact makes a great early warning system when things start to go wrong. Only a full internet diagnostic will do (I believe the PS4 can do that). Start with a reset of your router and then clear your system's cache. Look up any error codes you receive on Bungie.net or the Destiny app to bring up a help guide.