From your symptoms it seems like you have low power, The test I suggested will only rule out a bad cable or splitter port. If you can find the other splitter check all the connections (tighten them) and terminate any unused ones.
When you tried an amplifier where did you put it? Was that amplifier a 2 way design it needs to be to allow the modem signal to flow out (upstream).
On your modem do you have an upstream or downstream, neither, both. (look at the LED's on the front of the modem)?
Try these things and let me know what happens!!