A typical assumption in the auction literature is that bidders can submit any bid within the non-negative part of the real line. Yet, when auctions are used the set of bids is restricted at least to a countably infinite set due to the discrete nature of currency, and often even to a finite set. We study the effects of explicitly modeling a restricted set of allowed bids in a Vickrey auction within the independent values framework. We present a characterization of all pure strategy equilibria of the unrestricted Vickrey auction, obtained by Blume and Heidhues (2004), and analyze how this set changes when we restrict ourselves to the Vickrey auction with a restricted set of bids, although we do not provide a complete description of this set. We discuss the consequences of the change of the equilibria on the seller's revenue and the consequences of a set of equilibria that is present in both models. We examine a tool that is available for the auctioneer to obtain a positive ex- pected revenue, namely a proper specification ex-ante of a dictatorial tie-breaking rule. We also explore the consequences of introducing of a reservation price in our setup, knowing that it yields a unique equilibrium in the non-restricted allowed bids case, as shown by Blume and Heidhues (2001). Finally, we summarize our contributions and provide a list of the several questions that remain open in the field.