If one or even two (but not 3) of the four hanging Tzizit öéöéú threads
on one side of the knots tears off, partially or completely, the
Tzizit öéöéú are still valid, provided they were attached in the manner
used today, whereby one carefully ensures that each string is
divided on two sides of the knots. If one string from each side of
the knot tears off, the Tzizit öéöéú are invalid because we fear they are
from the same string. (MB 12)

****Adopted from the shiurim of Ha Rav Leible Sternberg