On a site where users can choose their own names, sometimes users choose names that are not, or are might not be, their own. If the name is common -- if Amy Adams is using the name Joe Smith, for instance -- it doesn't matter: nobody would have any reason to believe that the account belongs to any particular Joe Smith, of which there are many, and nothing Amy does is going to reflect negatively on any particular Joe Smith.
If the name is rare or effectively unique, though, and that person could conceivably be a user of the site, then we care a lot more. For example, if a user named Jon Skeet shows up on a programming site, or a user named J. Michael Straczynski shows up on a science-fiction site, and that's not really who they are, we don't want those users to use those names because they're misleading.
Then there is the vast space in between these two endpoints. I recently encountered a user who is probably not who he says he is. The name isn't famous and it isn't "Joe Smith" class, but Googling the name leads to a single individual with plausible interest in the topic of my site. There is, therefore, some possibility that what this user posts could be misattributed to an innocent bystander who would (in this case) be unhappy about that misattribution.
On the one hand, we don't want to require users to prove who they are; Google+ skirted that line with their "real names" policy in the early days and there was a lot of backlash. On the other hand, we want to be good Internet citizens, and we care about not harming people. It feels like we need to take this one case by case.
In a situation like this, what are reasonable (and ideally tested) next steps to take?