The bandwidth of a network card would generally outclass your internet connection by an order of magnitude.
Gigabits per second network cards have been standard for at least a decade.
And then there’s your wifi too, if applicable, but even that is much faster than your general internet connection, unless you’re far from your router. A new laptop might help you get a better wifi signal, if it had a newer wifi protocol or stronger radio, but that’s also going to depend on your router.
But in front of all of that is going to be your modem. Depending on the quality of your modem that might be a bottleneck.
Is it possible the hard drive is failing in the old one? I don’t know what the operating system would do if download speed surpassed write speed, but once it couldn’t hold it in ram any longer waiting to write I guess it would have to throttle it.