What version of Premiere Elements do you have and on what computer operating system is it running?
I appreciate that folks want the details but I left them out because this applies to several versions on several machines. Particularly v10 on windows 7 Pro 32 and v11 on windows 7 Pro 64. But the same thing applies on Mac Snow Leopard and Lion, on Win XP and on 32 bit versions. Every computer, every time I open Premiere Elements, it wants me to log into an Adobe account. I think that this is because some people share or store online. I do not.
The answers are in the details. It would be helpful to know what version(s) of Premiere Elements is(are) involved and if this is happening in a home or business or school environment.
But with the information supplied this would be my observations.
I have several versions of Premiere Elements on the same Windows computers (ranging from version 2 to 11), and, rarely if ever, am I asked to enter my Adobe ID and password. Consequently I find it unusual that every time you open a version of Premiere Elements on your many computer(s) you are asked for this information, emphasis on every time and many computers.
One guess may be that the computer - program recognition is being disrupted for some reason.
You could go to the Adobe web site to "My Adobe" account information and see what is going on with your account. You will find a lot of details there. Or, you make an inquiry with the Adobe Chat line.