sounds like a failure to create the authentication cookie, login succeeds but the cookie is not created.
only a few things I know that cause that
1. having the site inside a frame or frameset with a different domain in the outer frame. web browsers will not let one domain see the cookies of the other domain. some hosts have weird setups like that especially things like "domain masking or forwarding" often do things like that.
2. software on the client machine could also block a site from setting a cookie, any security or privacy guard type thing or even some web browser toolbar plugin can be at fault.
the first cause would happen from any client machine, the second one can be ruled out by trying to login from another client machine on a different network or at least without the same software installed.
3. another possibility is if you have the settings configured so that cookies are only transferred when using ssl/https, then if you go to a page without https the cookie will not be sent in the request headers therefore not seen by the site. if you have an ssl certificate you should check the box in site settings to use ssl on all pages. if you don't even have an ssl certificate you can rule out this cause