If you have a c onversion problem where the T4 is converted into Revers T3 (RT3) instead of Free T3. That could explain why the TSH goes up with increased T4 dosage.
I would recommend not only getting tested for Free T4 and Free T3, but also REVERSE T3. Many Dr's will not want to run this test. Not very common and as I understand it pretty expensive. But given your response to increased T4 medication may make it warranted.
If you do have a conversion issue, most often this may mean you may be best served and treated with a combination of both T4 and T3. Again not all Dr's are comfortable with this. You can get this combo treatment with either going to natural dissected thyroid medications such as Armour, or by the addition of synthetic T3 (cytomel).
What's more important than your TSH, which can be affected by many factors other than thyroid hormone levels, is what your FREE T3 and FREE T4 have been doing during this time. Do you have a labs history to post? If so, please include reference ranges as they vary lab to lab and have to come from your own lab report.
That is a little scary, although due to the slow acting nature of T4 (Synthroid), in theory, it isn't impossible...very unusual (did I say "very"?), however. Do you know WHY they suggested that?