Yes, it should. We definitely have tested that, but will do some more testing. With the "Facebook User Only" setting, and a brand new user for both, if you click the FB button on that page, we've even tested that it will create you a new Joomla user and then link that to both Facebook and LinkedIn. Always possible we missed something though with existing users, or something in general.
One thing I guess to test is the order of your JFBC and JLinked system plugins. Try swapping the order so the 'other' fires first. That may be part of the problem. If so, let me know.. that shouldn't be the case, but I can see it after talking through it.
Thanks,
Alex