Yes I can, as I have a wireless router at home set up with WPA encryption. My PSP connects fine with it, in fact it shows webpages of DCEmu faster than my stupid computer can (surprising, since it's quite a powerful one).
Maybe you should be closer to the access point, I sometimes get that bit of where it connects, exchanges keys and goes back to the first like you said because I'm not near enough and the signal is weaker.
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