The local area connection should always display as "connected" as long as your modem/router is turned on and your ethernet cable is plugged in at both ends. But doesn't necessarily mean your connected to the internet, as the local area connection represents the connectivity between your network device and your pc.
If your having to power cycle your modem to fix the problem it could indicate an issue with the modem, what you should check is if any lights on the modem are off before you power cycle, such as the adsl light or internet/ppp light if your modem has one of those.
I agree with dets00b. This is often caused by misconfigured modem or modem firmware bug. Also Dynalink RTA1320 is prone to internal capacitors failing which causes this problem.
If you have an RTA1320 try replacing it. Make sure the modem has the latest firmware version. Also make sure the modem is configured to auto-reconnect.
Also if you have another router connected behind the modem and both are configured for NAT, then you will see this issue. Under these circumstances you need to configure the modem for "IP extension" in order for the Internet connectivity to self-restore.
I think I have found the problem now why I do not connect . To over come this problem I click on start then run type in CMD when the next widow come up I type in NETSH space IP space RESET space RESET.LOG now push enter then reboot. Since I have done this the problem seems to have been fixed.