We study the complexity of bidding optimally in one-shot combinatorial auction mechanisms. Specifically, we consider the two most-commonly used payment rules: first-price and VCG-nearest. Prior work has largely assumed that bidders only submit bids on their bundles of interest. However, we show the surprising result that a single-minded bidder may lose an exponential amount of utility by playing his optimal simple strategy (only bidding on his bundle of interest) compared to playing his optimal complex strategy (which involves bidding on an exponential number of bundles). Our work suggests that it is important for future research on combinatorial auctions to fully take these effects into account.