This is taken from the BT Mate webpage. It doesn't list the Note 3 as supported, although I don't recall having Bluetooth issues with it in the past...
Q: What other Androids are supported besides the Samsung Galaxy S5 and Samsung Galaxy Tab4 & TabS ?
A: GarageMate requires Bluetooth 4.0. While late model Samsungs devices provide a flawless experience, the same can not be said with certainty for other Androids.
In particular Androids released prior to Fall 2013 (such as Motorola Droid Razr or any Nexus made by Asus), suddenly stops communicating with the receiver and require you to pair again.
You can help us by voting-up/starring the Bluetooth4.0 bug reports to Google. Please visit this Google link and this link to Google. We regret these issues and hopefully they will be fixed in the next Android release.