No, although poorly-written C programs might have.

The tm_year field of struct tm holds the value of the year minus 1900; this field therefore contains the value 100 for the year 2000. Code that uses tm_year correctly (by adding or subtracting 1900 when converting to or from human-readable 4-digit year representations) has no problems at the turn of the millennium. Any code that used tm_year incorrectly, however, such as by using it directly as a human-readable 2-digit year, or setting it from a 4-digit year with code like

tm.tm_year = yyyy % 100; /* WRONG */

or printing it as an allegedly human-readable 4-digit year with code like

printf("19%d", tm.tm_year); /* WRONG */

would have had grave y2k problems indeed.

(The y2k problem is now mostly old history; all we have left to do is fix all the 32-bit time_t problems by 2038...)