It does if the new units surpass demand, even in the short term. If there are 5k available units but there is demand for 15k over the next 90 days, of course building 1500 won't affect prices much, if at all. You need to build until there is some vacancy, then landlords are forced to lower rent to prevent their units from being vacant, unless they have the means to hold out for someone who really wants that particular location.
Supplying more of something always reduces the price of that thing. The thing to keep in mind, though, is that it doesn’t reduce the price that people actually pay; it reduces the price that people would have paid.
