We analyze the interplay between license auctions and market structure in a model with several incumbents and several potential entrants. The focus is on the competitiveness induced by the number of auctioned licenses. Under plausible conditions, we show that auctioning more licenses need not result in a more competitive final outcome, contrary to what common sense suggests. This is due to the nature of competition among incumbents, which sometimes exhibits free-riding. We illustrate some results with examples drawn from the recent European license-auctions for third generation (3G) mobile telephony.