I guess you're looking for:
if [ "$PHONE_TYPE" != "NORTEL" ] && [ "$PHONE_TYPE" != "NEC" ] &&
[ "$PHONE_TYPE" != "CISCO" ]
The rules for these equivalents are called De Morgan's laws and in your case meant:
not(A || B || C) => not(A) && not(B) && not (C)
Note the change in the boolean operator or and and.
Whereas you tried to do:
not(A || B || C) => not(A) || not(B) || not(C)
Which obviously doesn't work.