Technically it could, but in most real world situations won't. Most home equipment has more than enough bandwidth to stream local video content, except for low quality wireless connections. If you did somehow manage to saturate your local network, you would certainly experience issues. Even watching 4k or Blu-ray content won't come anywhere near the capacity of a home network, although it may exceed the individual bandwidth your device is receiving from your wireless access point. About the worst scenario I would imagine would be a cheap wireless router providing poor performance to your individual local video stream, or possibly becoming unstable and taking it's network offline until the router is rebooted.
TMTOWTSAC <= Most network equipment provides a full speed connection to each device. So, each port on a 100 mbps router is going to get a full 100 mbps to the router, not a portion, and internally routers usually have even more bandwidth. What you are describing sounds like the hub devices that haven't been used in quite some time, in which every device was sharing common bandwidth.