Interesting question. I'd say it will probably work in your special case, but it will not work, in general. Here are the factors I'd consider.
Costs vs. Expenses Effect:
Several sites use a "donate-what-you-like" model which is quite similar to a "pay-what-you-like" model. Wikipedia, for example, does it twice a year. It seems to work but I think there's a very important factor: These campaigns emphasize expenses. However, every business guy knows that's not the same as costs.
Unfortunately, people seem to make a distinction between these two cases. Asking for money to cover your expenses seems acceptable; asking for money to cover costs seems less acceptable.
For example, the developer of Ardour, the leading Digital Audio Workstation for Linux, started to ask for donations to cover his costs of living back in 02/2009. There's no history but I think he didn't make reach the requested amount of US$4500 at least in March. He also needed to employ additional "tricks" to make people pay: For example, offering the option to sponsor issues, and a friendly reminder to Apple users during the download, etc.
Another example is the GNOME desktop project. There's also a donation campaign, called Friends of GNOME. One thing that seems to pop up whenever the program is discussed elsewhere, is the income of the director. People write things like "I'm not willing to donate because her income is way to high."
Another way to put it: People seem to be willing to help you prevent losses. But they are less willing to help you make an income (or profit).
Free Rider Crowding Out Effect:
There will always be free riders. Essential for success, however, is to never let people know how many free-riders exists. Otherwise, their example will crowd out the Payers. This is basically the negative effect of social proof: "If the majority doesn't pay, why should I?"
Unfortunately, this sort of information will always spread. After all, companies compete and one way to do so is by lowering prices. That requires to lower costs, so they will stop paying the pay-what-you-like services, first. When the information is shared, people will stop paying.
The Dog-Bites-Man Effect:
Today, these sort of models make headlines, especially because the Intellectual Property/Piracy discussion is in full swing. Thus, Radiohead and the World of Goo owners have no problems raising attention and money. It's exceptional. However, the more people or companies use such a model, the more common it will become. The amount of free riders will increase.
Overall, these reasons will make a pay-what-you-like model unfeasible for businesses, in general. But for keeping a small web site running, it could work.
Hope this helps.