My Logitech Bluetooth mouse was notably flakey yesterday, not working worth a damn after sitting idle for even a few seconds. I poked around online and found that turning off Handshake can make the problems go away:
If your Mac’s Bluetooth connectivity to peripherals is flakey, it’s worth a try, only takes a few seconds, and doesn’t require any tinkering or messing around with system files. Simply open up your System Preferences, select General, look for a setting called Allow Handoff between this Mac and your iCloud devices and make sure it’s un-checked.