Ther final output from a traceroute looks like this:
15 274 ms 273 ms 274 ms 58.71.0.82
16 274 ms 273 ms 273 ms 124.83.5.54.pldt.net [124.83.5.54]
17 273 ms 275 ms 272 ms 203.111.226.14
18 302 ms 299 ms 300 ms 203.111.239.206
19 299 ms 299 ms 300 ms 203.111.239.205
20 301 ms 300 ms 301 ms 121.1.3.149
21 303 ms 306 ms 302 ms 121.1.6.155
22 307 ms 307 ms 313 ms 121.54.110.74
There's two notable fields about this string. One is that most of the IPs do not have DNS. The second is that some of the IPs are nearly identical and on the same router subnet.
By putting both those pieces of information together I would venture to guess the user is right, sort of. They are all probably in a small, managed region and share one IP. This region could be a large office building, a town or even a small city depending on bandwidth. Even better, some free wireless or free dialup services can make everyone look like they are from one IP.
That doesn't mean that they aren't all one person, but it's possible that they aren't totally lying either.