This will depend on the capacity of the iPhone. This will also depend on the size of each picture which will depend on their format, quality and resolution. A 16 GB iPhone for example can hold around 16000 photos.
I have had around 3000 pictures (about 10 videos) on mine at one time, before I backed them up to snap sync and deleted them off of my phone. My phone still let me take pictures and video, and it was not running slow.