I have recently seen a Chasid wearing two baatim (Rashi and Rabeeinu Tam) on his head but only (Rashi) for Tefillin Shel Yad. Later, a third Tefillin Shel Rosh were applied but the Shel Yad stayed put. Is this halachically correct?
This practice is halachicly justifiable, although I've never heard of people doing this in practice.
The Shulchan Aruch when bringing the machlokes about the order of the parshios brings from the Rosh and other Ashkinazi poskim to have two tefilin shel yad, one in each order. However, it is not so clear that the order of the shel yad changes according to Rabainu Tam. The Rambam in a teauvah (brought by the kesef Mishnah maybe?) brings the order of the shel yad as a proof to the rashi order against RT.
A third order for the parshios is the Shimusha Raba, which is the reverse of Rashi. There is an opinion, I believe the Ravad, that is the reverse of RT, but the Arizal was machmir for SR and not this fourth order. In the shel yad, the SR holds like Rashi, and the fourth order can hold like RT, which as I said is uncertain. The Ari put on SR tefilin at mincha, but why not after shacharis?
Some have the custom of wearing both at the same time. The ou site has a discussion concerning Rashi and Rabbeinu Tam tefillin. See point 4.