Google account sign in problems
I've the exact same issue on a Galaxy Tab P1000 with gingerbread 2.3.3 since about 2-3 weeks.
My current interpretation is that Google has updated its certs (the current one I get is dated 11/7) and the new ones are not trusted by Android anymore making it impossible to use any google account for mail/play store/etc... If this is the case, the only solution would be an update from the manufacturer or maybe rooting the device and convincing it that the new cert is good.