Let's assume everything,
Are you connected to the internet, data on, or wifi connected with the correct password?
Have you recently changed passwords for accounts that won't sync?
Do you have global sync set to on, and/or do you have settings for frequency of sync? i.e. every 4 hours, 15 mins, etc.
Can you access the Play Store? Does it log you in or do you get an error?
Has your device updated OS lately?
Are you on a restricted network? WiFi from your employer or other private networks may block certian information from being accessed or updated.
Do you still have service on the device? Without service your device may still show a data signal but will not allow access to network data so you will be forced to use a WiFi connection.
Does your phone show you are connected to a network or WiFi? Are you in airplane mode or have data switched off in your settings?
Do your apps that require network work? If you open an app like Facebook or Gmail will it update when open, or does it continue to show you old data from the last time you were able to connect?
Look at these things and post back.