I think it depends what your standards are. I've lived close to NYC (doesn't typically get much colder than 20ºF / -7ºC, rarely more than a few inches / 8cm snow). I've also lived in Rochester, NY which is consistently ranked in the top-10 snowiest cities in the U.S.
Bad news:
Madison gets cold, but not as snowy as I've dealt with. We've had a few "polar vortices" in the last couple years, where the temperatures can dip to -40ºF/ºC (although they only last a few days before temperatures return to normal). It's not unusual for the temperatures to be in the 10's-ºF / -9ºC for a few months. There's also a decent amount of snowfall, and it's not unusual to have a few storms that can drop half a foot (15cm) up to 1.5 feet (45cm) in one day. Winters are pretty long... expect chilly weather starting in late September / mid October. Expect cold weather and snow from November to May (there will be a few nice days before May, but don't let it fool you).
Good news:
When my parents and friends express horror at those numbers, I like to tell them that it's not as bad as it sounds. Once the temperature gets below about 20ºF/-6ºC, you're bundled up so warmly that you can barely even notice the difference between that and -40º unless you're strolling around outside (which I advise strongly against). Plus, the winters here are much less depressing than the ones in, say, Rochester. The sun shines in the winter, even when the temperatures are low. That makes a huge difference to peoples' moods, and makes the countryside gorgeous. Many, many residences and employers in the area provide indoor parking, so you barely ever need to be outside (unless you take the bus, or work outside, or have another reason to be out there). The snow isn't too bad, because the city has a fleet of plows and salt trucks to maintain the roads even in the heaviest of snowfalls. It's a bigger problem if you have to shovel your own driveway, though. There are good snowboarding / cross country skiing opportunities in the area. The summers are gorgeous if you can survive the winter.
In summary, if you don't mind the winter too much as it is, it won't be that bad. Learn to layer (e.g. wear a few lighter layers instead of a single heavy coat). Make indoor/underground parking a priority when you look for a place to live and work. Make sure your car isn't bad in the snow, and alternatively buy good snow tires before the winter hits. Do those things, and you'll be just fine 