use this as a starter

(Cos[27 W t] + 3 Cos[W t] + Cos[15 W t] + 4 Cos[28 W t]) /. 
 Cos[x_  W t] :> 0 /; Not[MemberQ[{1, 29, 15, 31}, x]]

You were counting on the fact that Mathematica puts the numbers first and only then the symbols. So, to be more general, check that x is actually a number

(Cos[27 W t] + 3 Cos[W t] + Cos[15 W t] + 4 Cos[28 W t]) /. 
Cos[x_  y_] :> 0 /; NumberQ[x] && Not[MemberQ[{1, 29, 15, 31}, x]]

or you can check that x is a number inside the expression

(Cos[27 W t] + 3 Cos[W t] + Cos[15 W t] + 4 Cos[28 W t]) /. 
 Cos[x_?NumberQ  y_] :> 0 /; Not[MemberQ[{1, 29, 15, 31}, x]]

(The ? is called PatternTest) and the /; is called condition.