Welcome to Inkscape Community!
Unfortunately, Inkscape doesn't allow more than 2 path segments to be associated with each node. However, there is a way to sort of make it work like there are more than 2.
First, you need to start the new path precisely on top of the existing node. Usually Snapping is the fastest way to do that (manual info on Snapping:
http://tavmjong.free.fr/INKSCAPE/MANUAL/html/Snapping.html). But you could also use the x,y coordinates on the Node tool control bar (copy them from the other node and paste them in for the new node, after you finish drawing the path).
Now that they are on top of each other, you need to select them both at the same time, whenever you're ready to move them. You can do that by dragging a tiny selection box around them, with the Node tool. That way, they are all moving together, as if there was only one node.
Let us know if that helps