After upgrading to SourceTree 1.6.24, BitBucket Cloud and BitBucket Server authentication fails with saved credentials and entering the correct password does not get accepted either.
Downgrade and install SourceTree 1.6.23 (or an earlier version): http://downloads.atlassian.com/software/sourcetree/windows/SourceTreeSetup_1.6.23.exe.
As a temporary workaround copying the attached Askpass.exe.config file in to
C:\Program Files (x86)\Atlassian\SourceTree (or the installation folder respectively).
This will provide the necessary configuration and should fix the multiple authentication requests.