I think sometimes people say things that come across that way in order to offer potentially helpful advice, not to one-up you. For example, if you had a minor house fire recently but my house had burned completely down at some point then I might tell you my story as both a way to let you know that I understand how difficult your situation is and offer assistance with what comes next. Dealing with insurance and replacing your stuff can be an overwhelming proposition. If I don't communicate the desire to help you particularly well then you may think I'm just trying to say that I had it worse when really I'm wanting to help you but am simply too awkward to do it effectively.

Obviously some people are trying to shift the focus off you and on to them but I think you should be able to tell someone's intentions with a quick follow up question to whatever they say. Something aimed at determining whether they want to help you or just talk more about them.