This is NOT a bug, not a server (vanilla or bukkit etc) defect, it is deliberate and intentional. The server you're trying to connect to must be in offline mode. As of 1.7.9, in preperation for 1.8 and the new naming system (UUID instead of username) they ALSO apparently are preventing offline mode servers from using the skins. I can see the point, to an extent, but sadly, from personal experience many servers run in offline mode for testing plugins, which as they've been churning out Minecraft updates left and right, sadly has left many constantly testing and working with dev's of plugins to get things going for the new version, etc.
You are a lifesaver. I have been all over the internet since 1.7.9 came out, trying to figure out just what the heck happened to skins. This is exactly what happened. Nasty thing to do to all the people running in offline mode. I haven't seen any problems with the changeover to online mode yet, except for it wiping my inventory.
Is there a specific reason that I'm not seeing behind Mojang changing this?