Yes, definitely. I'm an academic and work within the university system. And where you get your degree definitely matters, but it may matter more or less depending on your field and the geographic area you're wanting to work in. So in some places, having a degree from a local university that has a strong alumnae network in the area can be incredibly beneficial if you plan to live in that area. This is the case with a lot of 'state' universities in the U.S. So if you went to the University of Florida, UF alums have a really strong sense of identity and strong local alumnae networks. You might do better in these kinds of areas if you have a degree from a certain loved local university, even when up against folks from more prestigious out of state schools. There are also certain universities that are particularly well-respected in certain fields, and they are generally good universities overall, but they aren't necessary your Havards and Yales and Oxfords. The University of Chicago is very well respected in my field, which is Sociology. If you get your degree from them, rather than say, from Yale (which doesn't have the greatest program), you'll be much better positioned for jobs in that field. My graduate university was the University of California, San Francisco. Most people don't even know there is a UC in San Francisco, but it's the top-ranked medical school in the country. If you want to go into medicine, it would benefit you a lot to graduate from there, but it's not Harvard), though Harvard has a good (just not as highly ranked) medical school as well.
That said, many of the top universities have very strong alumnae recruitment networks, especially in particular fields and in particular cities. So your Harvard's/Yale's and your Oxford's/Cambridge's are really good at linking students and alums to jobs via their alumnae networks, and their particularly good at this in the geographic areas where their former students tend to end up. So for Harvard and Yale, this would be Boston, New York, Washington DC, and for Oxford/Cambridge, in London. If you have a degree from one of these universities and you can be connected through their alumnae network to employers, who are also alums, and who are hiring, you will likely be favoured for a job over others who don't have those connections. But that might not be beneficial in every geographic area or in every field.
Certainly, networking can play a huge role in pursuing a career and having a connection like attending the same university can open up doors in networking for you, but it's not the only thing and it may not always be the most important thing. In my field, my university is particularly well known for the type of sociology I do and the type of research methodology I work with, so I have that up and also that additional hands-on training in those methods that other folks in my field, who might have gone to Harvard or Yale, wouldn't have. So that's a big bonus. But marketing myself and my work (publishing, presenting at conferences, making connections with others in my field) is a big part of it too, and that isn't really dependent on what university I went to. It's much more about personality and motivation. So yes, it definitely helps, depending on where you want to work and in what field, but it's not everything. If you have a choice between a big name school that isn't quite the right fit or won't offer you as much support or enable you to really do what you want, and a lessor known one that is a great fit and will let you study exactly what you really want to do, I'd take the lessor known school. Certainly, when I have done hiring, someone from a prestigious school might have stood out, but only if they had the experience and something special to offer to back it up. I'm much more interested in drive and talent than credentials and there are lots of ways to demonstrate that in the interview process other than just having a flashy degree.