Can you try the following:
(1) Start -> enter "network connections" (without surrounding quotes) in the search field, press ENTER; this will produce the Control Panel -> Network and Internet -> Network connections dialog window
(2) select the network connection for "local area connection" (i.e. your wired connection that is failing to start properly), and then right-click on it and select the DISABLE item, then wait a few seconds until a red "X" appears over the network icon in the system tray notification area
(3) right-click again on the selected "local area connection" network connection icon, and this time select the ENABLE item, then wait a few seconds after which hopefully the red "X" on the network icon in the system tray will disappear and the network icon will now show as clean. Your network connection should now be working.
You should now be able to get to the Internet.
No explanation as to why this doesn't happen automatically at boot time (other than a screwup of some kind in the driver installation, which unfortunately is unlikely to be "fixable" in my experience without the extreme solution of a clean Windows reinstall), but the DISABLE/ENABLE workaround should at least get you going.
This happens to me as well, on my Lenovo W530 laptop, with its wired connection to my router failing exactly this same way at boot time... inexplicably... but "cured" every time with a quick DISABLE/ENABLE. I don't know why this happened or when, and it absolutely was NOT the case initially back in mid-January when I freshly reinstalled Win7 when swapping a new SSD for the SATA spinner in the laptop. But sometime after that it crept in, and nothing I've tried will fix it.
So I have given up, and decided to simply live with the 100% reliable DISABLE/ENABLE solution/workaround. I'm not about to reinstall Windows again just for this problem.