It is possible to use a DDOS attack to gain information. In addition to the answers by Rory Alsop and H. Idden which focus on gaining infrastructure information or overloading the incident response team so other attacks stay undetected longer, there are a few other possibilities.
Auto Scale Applications - When working with an application platform that auto scales a DDOS attack may use the fact that they auto scale to do something. This would have to be in concert with some other kind of attack or information, but the general idea is that if you know enough to exploit the "reboot" process you could use DDOS to force a new server online witch would go through the reboot process, allowing your exploit. It's important to note that this is in addition to an already existing security problem. It's just the tool by which the totally different exploit is brought online (or maybe tested). A sample I can think of is a Production server hosting it's code base in a public github repo, when scales pulls new code into the server, then starts. You could add bad code to that github repo (if not managed correctly), force a restart and have your exploit.
First come First served applications - There are some applications that are "first come first serve" in some part of their process. IRC (from comments) is an example but there are others. Any service that reserves something for a user on a first come first served approach, can be exploited via DDOS. Either by taking up all the slots till a specific person get's theirs, or by forcing a reboot and getting anther chance at "slots" after the restart. These are pretty common, and while a service that uses this for authentication is a bit odd, it's not unheard of. In fact licencing services do this all the time. First user with a licence of "ABC" is the qualified user of ABC. If that data is lost after a reboot then DDOS could cause that reboot, and get their foot in the door.
Startup Vulnerabilities - I have seen, quite a few times where a server boot resulted in services being "left on" just in case. For example , let's leave SSH passwords on after reboot then turn them off after a couple of hours just in case we need emergency access. Or, FTP is on for 30 mins after reboot. This is more common on network appliances. Things like "insecure wifi access for the first 2 mins" or "anything can upload anything for 2 mins". Usually this is part of an upgrade/update mechanism. For example a Cisco router may accept any TFTP data it finds after reboot. Some computers will listen for ANY netboot server at restart. DDOS can kick off the restart and allow your "bad code" a way in.
In essence, a DDOS by it's self can tell you some things, but usually only stuff that is useless on it's own. A DDOS in conjunction with other attack vectors can be extremely effective.
Note The methods used here would work in a lab, but there low hanging fruit for a security team (or even just regular IT). While they do exist in the wild an attacker would have to have gained access/knowledge of internals way beyond that of someone just doing a DDOS.