The fax machine is not receiving faxes. It shows that the line is in use, but no faxes are coming through.

Yes, The fax header should show you who sent the fax to you. Every fax is supposed to have a terminal ID (senders fax number and Name)

Andrew

Your fax machine thinks everything is OK. If you can receive you can send.

call the failing fax number with a regular phone and listen for fax tones.

It could be some senders are sending over a DIGITAL PHONE LINE. All fax machines are ANALOG.

Problem lies in your Receive Mode setting. If you don't have a dedicated phone line for the fax machine, you need to set it up for either Manual where you answer every call or Fax/Tel whereby the Fax machine determines if the incoming call is a fax or phone call. Instructions per Intellifax 4100 Manual.

Select a Receive Mode
Choose the mode that best suits your needs by pressing the Mode button
repeatedly until the FAX and FAX/TEL lights show the setting you want to use.
Manual
You must answer every call yourself.
The Brother machine must be started manually
to receive faxes.
Fax
For use with a dedicated fax line.
Every call is answered as a fax.
Fax/Tel
For receiving fax and voice calls on one line.
The FAX answers every call and receives a fax or
double rings for you to pick up a voice call.
TAD
For use with an answering device.
The TAD answers every call and stores voice messages.
Fax messages are printed.

Problems receiving could be any number of problems. The first item I would check it to make sure no other devices are on the same line as the fax modem, such as a fax machine. This can cause a problem for a fax modem to receive faxes when another line is being picked up and listened to at the same time.
http://www.getfaxing.com/forums

Make sure you are plugged into the LINE port of the fax machine.

First, check your fax's phone line. Put a regular phone on it and check for dial tone and line noise. If the line is dead or noisy, this will cause a problem. 
You can also try dialing your fax number from another phone and see if the fax unit picks up the line and starts initiating communication. (sounds like modem noises)

If you can copy ok you have effectively ruled out your machine as you have used both the scanning and printing mechanism of the machine.

It will probably be the fax on the other end as if it sends a poor quality page that's all your machine can receive, a poor line can also cause problems if it is from multiple faxes. If so have your provider check the line (you probably hear static on the line).

MAKE SURE YOUR PHONE LINE IS PLUGED INTO THE JACK ON THE FAX THAT IS MARKED LINE ALSO CHECK THAT THE MODE KEY IS SET TO FAX