The game has not ever loaded correctly? It always disconnects after the first loading screen?
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Try unplugging your router and modem for ~5 minutes and replug them.
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If you have antivirus/internet security/extra firewall software (eset, norton, mcafee, etc). As a test you may try disabling them temporarily and see if it solves it. If it does, re-enable the program and add exceptions for PvZ:GW2. Software like Peerblock would also block the connection likely.
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You may try uninstalling and reinstalling Origin. I doubt this is it judging by the message but it's worth mentioning.
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These two links may help:
http://help.ea.com/en/article/plants-vs-zombies-gw2-known-issues/ (see the Port Forwarding section, an easy way to test if this is the problem would be to add your PC's local IP to DMZ temporarily, if it works then it's likely a port forwarding issue, I would not recommend leaving your PC in DMZ outside of just testing)
http://help.ea.com/en/article/connection-troubleshooting-basic/ (see the sections pertaining to PC: uPnP, network discovery, IPv4 and testing with UO trace//pathping//traceroute)
http://help.ea.com/en/article/connection-troubleshooting-advanced-pc/
