The same happens on my system (Stretch) and it's not just the mouse and keyboard but other devices, too. For example, I can never connect more than one bluetooth gamepad at the same time - doing so simply disconnects the previously attached one. Connecting only one pad always disconnects the keyboard for a couple of seconds, too.
However, after trying different combinations I have found a workaround that seems to isolate the mouse as the culprit, at least on my system: When it is disconnected, neither the keyboard nor the gamepads (tested up to four) exhibit the behavior described above. Connecting the mouse after connecting the keyboard / pads doesn't disconnect them either. I figured that, though it has worked on my previous and current laptop until I installed an Intel 7260NGW M.2 card (in lieu of a BCM4352), it might be the mouse, but replacing it with another model yielded the exact same results. The behavior seems to be (in part at least) dependent on the wireless / bluetooth NIC. Any ideas where I might start to look to solve this? I can post any logs / details you might need, so please let me know what you need.
