My one word for a solution to that many faxes, assuming you have DID, is RightFax (RF). I have been using it over 6 years with excellent results and I am the phone guy for a hospital with 6 other connected sites plus a few more. With RightFax you have no more faxes mying in trays or copiers waiting for pickup and there for anyone to look at - everything goes in a mailbox. For outgoing, we are saving pallets of paper a year because the documents they usually print to paper and stick in a fax machine get printed right to RF, those at the "front desk" who have scanners now can scan directly in, and others go to the copier and scan to their E-mail and just attach the resulting file or high volume departments have their own scanner with a document feeder. We do not give everyone their own fax number because we have too many people, but instead assign one to three (depending on what they have now or department groups) mailboxes per department. In our case there is a central server that has a T1 board that conencts to a T1 on the PBX and provides 23 "fax lines", and then client software that goes on each PC that needs access. We use ISDN PRI for our primary trunking. It also supports many other connection methods and protocols, as well as direct connection to MS Exchange and web-based access. They even have software modules to interact with maybe 20 different kinds of copiers so you can actually send out right from the copier over the network. We also have AVST CX-E for voicemail and Unified Messaging and RF can deliver messages to those mailboxes and also you can have your fax calls come in on your office extension and it will route them to RF. In RightFax for Fax status it will tell you if it was successful, or if it wasn't it will have error codes that once you have the "translation document"

P will tell you the most likely issue with a line, like "loop current drop", "transmission errors", etc...
Anyway, in relation to the actual question, I have had a variety of issues with analog and vendors for years, and I have a bit of an attitude... Part of it is because we use ISDN PRI trunking and there are some limits on the hardware in some cases. Back in the internet days here no one could connect faster than 28.8K on an outgoing call from an analog call - I was told it had something to do with how much the hardware can handle. I have had the same situation fax vendor blames my phone line, I point back at the fax vendor. The way I end the discussion is I plug an analog phone into the line, dial a digit to break dial tone, and if it's quiet it's not my line so go away. It could be a bad line cord or a bad channel on the PRI, it could be that if faxes are piled on only a few boards and they are busy a lot you could be generating more traffic than the board can handle well - try spreading them out, or if you put faxes on with a busy call center it could be the same way... It's almost impossible to troubleshoot without expensive equipment. Normally if you have bad call quality you can hear it in your ear. If you have any kind of call traffic analyzer you could make a report to monitor the fax lines and see if there is a common trunk or trunk channel that seems to have issues - put a butt set on each analog trunk and break dial tone to see if there is any noise on the trunk. If you ask the phone company to check them you will get a "no problem found" report most of the time.
In other news, I have had trouble with faxes using direct outside lines that are multiplexed using a "PairGain" device because the provider didn't have enough copper coming into the building for as many lines as you needed, and also fax machines that are behind an RJ31X jack for a security system because if the system dials out for a line test call it will unceremoniously drop your fax call. Lines run over the top of fluorescent fixtures can have induced noise as well as a few other induction sources that can be on the desk, including an IP phone.
Anyway - there's my nickle's worth

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Don Bruechert, Voice Comm Analyst II
CareTech Solutions @ Holy Family Memorial
Manitowoc, WI, USA