I’m sorry if I may not be posting this on the right thread.
My question is:

Is it possible to have too many branches to a landline where there isn’t enough power (signal) going to the telephones on the branches and if so what can or should be done about it?

The problem you're describing is too many extensions, or not. The limiting factor of landlines, assuming a regular copper line from the local telephone company, is ringing current. In the US, telcos are required to provide a minimum of 5 REN, if I remember right, which is the power to ring 5 standard 2500 type sets. Each phone should have a REN number printed on a label somewhere. It can be anywhere between 0.0, and 1. I've heard of phones with a REN number of higher than one, but never actually seen one. Simply add up the REN numbers from all the phones and if its more than the line can provide, you'll have to either unplug some phones or unwire the ringer in some to get that number down. Other alternative is ring boosters from a company like Mike Sandman Enterprises. The main problem with those is cost. If your landline comes from some kind of VoIP box or cable company box, the available REN could be as low as 2. I hope this was somewhat helpful.

To summarize, the limitation isn't really the number of phones, it's the number of ringers.

Chris, I'm not sure that is what the question is.
Need a better explanation of what the question is.

I think that Chris has the right answer to the question. Too many branches on a landline reads like too many extensions on a single phone line.

That's how I read it. Too many phones on a single line. Or, more accurately, how many is too many...

Assuming the OP has fairly modern phones, I doubt he has much to worry about. Most new phones with electronic ringers use something like .2 REN or something. Cordless phones are often 0.0 REN, since all it has to do is trigger the circuitry in the handsets. They're all battery powered anyways. The number of phones has no effect on the talking circuit, unless you have a bunch all off hook at once like an ad-hoc conference call. Otherwise, to the best of my knowledge the only limit to the number of phones (branches) on a single line is the available ringing current. If I'm talking crap, please- someone correct me.

You were very helpful.
Thank you very much.