I can't guess as to the multiple possible reasons (beyond ISP, or internal conflict, router issue, or blocked port issue, maybe) but I am reminded of this old thread...
http://www.ffxiah.com/forum/topic/35397/i-cannot-stay-long-on-ffxi/#2165045
...it discusses something similar. The solution for that user was actually the PlayOnline > Network Settings > (Enabled/Disable) Enable Connection setting, but the other possibilities (i.e. ISP/IP conflicts, wireless routers, router settings, blocked ports, etc.) are touched on. Hopefully you find a solution, also!