I had the same happen also. Left LW running overnight in the morning I had the message you do not appear to have an internet connection.

I closed the warning box, and checked LW. It was still d/l and u/l.
It was as though LW had just come out of idle mode. Only had 1 connection in the connection tab window, and showed it being connected for about 9+ hours. It was connecting to other as I was checking it.

I did not have to restart or reconnect LW.

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WindowsXP MCE sp2
P4 3.0 Ghz HT
4 Gb DDR2 @ 533 Mhz