Monotonicity and Threshold function
If I understand your question correctly the answer is yes. In the paper

"Majority Gates Vs. General Weighted Threshold Gates" (http://www.nada.kth.se/~johanh/majorityvsthreshold.pdf) by Goldman, Håstad, and Razborov

They show that anything that can be computed by a depth $d$ threshold circuit can be computed by a depth $d+1$ unweighted threshold circuit of poly size.

Your question seems to be a special case of this for $d=1$.