That can be a characteristic of cable faults, fritzy one moment fine the next.
It could be a break in the wire, or poor crimping at either end.
I suggested the cable as it is easy to replace, and cheap, and therefore a good starting point.
With regard to the modem, this is tricky considering the cost.
The fault could be a dry joint or a failing component.
I would consider buying/borrowing another ethernet card for testing purposes, unless of course your modem has a USB interface that you could try.
I would send you a card, I have several, but it would take too long to get to you!
A basic card can be bought for as little as $19.99 from Best Buy.
In an ideal world you would have access to a port tester that would help confirm correct operation from your PC up to (but not including) your modem, however some cables will report as OK on these testers yet fail when an attempt is made to pass data over them!