Studies that have ever been published or scrutinised outside razor companies seem to be rather difficult to come by.

The best I could find was Measurement of beard hair removal efficacy using image analysis, a study conducted by researchers at Gillette (kudos to Solveig_A on Reddit for fetching the full-text for me). Now, a warning: this study's primary focus was not actually about comparing two razors, but on techniques for finding statistically significant results even for marginal effects were to conduct such a study. That can make reading it a bit obtuse (I shall not get into why they might be motivated to conduct such a study...).

The first two experiments compare different types of shaving foam using a twin blade and a single-blade razor, respectively. They do not include results in such a way as to make direct comparison possible. However, the third experiment compared a single-blade and a twin-blade razor (Gillette's Trac I and Trac II razors, respectively) with the same type of shaving foam. According to their data, the twin-bladed razor was 1.6% more effective than the single-bladed razor, but that they could not reject the null hypothesis (that they were equally effective) at the α=0.05 significance level.

In short, this does not represent compelling evidence that a razor with two blades is more effective than a razor with one blade, which does not bode well for the claim that five blades are more effective than three.

Cecil Adams, writing for The Straight Dope, cites studies by Consumers' Research magazine which consistently failed to find a benefit for twin blades, whilst noting that the single-bladed razors tended to last for more shaves than twin-bladed razors.