Magic Mouse 2 says 'Bluetooth-enabled Mac computer with OS X v10.11 or later' so 'no.' Earlier versions should work on 10.5.8 and above.
There no real reason for bluetooth to stop working, other than innate diabolicalism, unless it's a hardware issue with either the mouse or computer. Do the usual PRAM/SMC resets for your machine and if that doesn't work, unpair and delete all devices. If that doesn't work, reset the bluetooth module (google for the instructions on your particular version of OSX).
Bluetooth does occasionally get flaky and, in my experience when it does it requires a reset. I had the same issue with this keyboard, it started spontaneously disconnecting, and more annoyingly sssssssssšhjjsasssssssłeqwrwqtyù for no reason, often creating a string of commands like close-this-app-and-no-don't-save-changes. Anyway, nuked it and reset everything and it's right as rain now.
If it is hardware, then USB Bluetooth dongles are cheap as chips these days and there's quite a few that claim Mac support at various OS levels.