An active open TCP will test the network conditions by measuring the responses from the server. If the routers buffers are saturated by another connection and causing high delay, the client will tune its send window accordingly. If a classic congestion control method is in play the client will continuously increase its send rate until a drop occurs. So it is possible for a new comer to muscle its way in if the device that is saturating the router begins experiencing severe drops while the new comer does not.
There are now experimental and more intelligent congestion control methods that use delay as a metric as opposed to simply using drops to gauge network conditions. However, these new methods have a bad habit of being bullied by the more tenacious classic methods that take as much bandwidth as they can.
Also with UDP connections if en masse, will dominate any network because of the careless spraying of bits it does. Depending on the application, it will recklessly saturate a routers buffers with no regard for network conditions.