The filter is to prevent the mix of data and voice signals, and one is needed for every socket where you have a phone / fax plugged in. (It's mostly to stop horrible screeching in your ear, but there are also some chances of audio interfering with the ADSL and causing your router to disconnect.)
It's quite a long time since I've looked at a Netgear router, but there may be some 'system log' display that can report what errors it detects, or the condition of the phone line.
Also, sorry if this is bad news, but years ago one of my clients had a problem with their Netgear router within a few days of purchase... what happened was that the "modem" part died, giving engaged tone for anyone dialling the phone number, while the router part was fine... so we could connect to the control panel and it said it was trying to connect but nothing ever happened. It's possible with any brand, and that was 8 years ago, but just a chance part of the Netgear has stopped working.
One other thing is that sometime phone wiring includes 3 wires to extensions. Over the years, I've seen recommendations to remove the third wire (called the 'bell wire' from the time when there were those old style phones with two bells under the part where the phone sat). We now have lots of smaller phones, some cordless and others which have no bells, so the wire is not needed. This third wire sometimes picks up mains hum and other electrical noise in the house and gives problems for the ADSL router.
Do try disconnecting extensions and have just the router plugged into the master socket. Power up the router without the phone line connected and then watch any lights on the router once you plug in the line. Many routers have one LED to show the line is connected and some have a second to show a connection to the ISP is established.
If you get no change when you plug the line into the router, there's a chance the connections are not being made properly (plug should lock into place - one of my many routers always seemed loose and the wire could fall out until I used sticky tape to be sure it was in!)
Hope it starts working again for you.
If not, it may help to know the model number for your router, in case someone else has had problems, and for checking the user manual online to see if that'll help.