MB 12: They are Kosher - Because they are long "enough to tie," but merely because of their thickness he cannot tie them together.
12:3. "Enough to tie" is according to Rashi from where the knots end, and the threads become free of each other. According to the Ri, even if the entire loose section has been cut off, and "enough to tie" only remains in the wrapped and knotted section, it is still Kosher. The common custom is to follow Rashi - and where it is impossible otherwise, (13) the Ri can be relied upon.
MB 13: The Ri can be relied upon - And it is permissible to wear the garment. Earlier in paragraph 1, where it was written that "where possible, it is good to be concerned for Rabbeinu Tam's opinion," the meaning was only where it is possible to easily find other Tzitzis. However, it is not necessary to go after them if they are not in front of the person, or in searching he will miss praying with the congregation. And here, where it is written "where it is impossible otherwise," the intent is in a place where Tzitzis cannot be found at all, then one can rely upon the Ri. Nonetheless, one should not make a blessing on these Tzitzis, and so too it is forbidden to go outside with them on Shabbos into a public area [where carrying is forbidden]. And in this case it is also acceptable, in an emergency, to accept even if some small part of the knotted section itself is cut. And know that all of this is irrelevant in a case where the string was cut where it passed through the hole in the garment, because there even according to the Ri it is invalid even in Torah law [vs. rendered invalid by the Rabbis]. And it is clear that where the Tzitzis have become invalid, it is necessary to remove the garment, and it does not help to stay standing where he is and not walk 4 footsteps, because the law of 4 footsteps applies only to going without Tzitzis [meaning, that one should not walk 4 footsteps without them]. Similarly, one should not fix the Tzitzis while the Tallis is still upon him, but should take it off and then fix it.
Yaakov Menken menken@israel.nysernet.org