The short answer is Yes.
The long answer is:
Applying for jobs and internships is affected by a few things. Opportunities, Connections, Uniqueness.
You can be unique in a variety of ways outside of coming from an Ivy League (or equivalent) school. Someone who is top of their class at Penn State is going to be more impressive to most employers than someone who was just average or worse at Dartmouth or Cornell, for instance.
"Opportunities" is a place where school matters, as the best companies will actively recruit from targeted schools. Here you want to go to schools that are the best at what you want to study, rather than just overall "name" of the school. Top state schools like Penn State and Univ. of Michigan are going to do well in this department too.
"Networking" is where an Ivy League-type school can help you. Small, elite, private schools are generally going to have closer alumni networks, and catching the attention of those alumni is easier when you are in a class of 1000, rather than a class of 10,000. Also, these these schools are considered elite because their alumni are in prestigious places, so the alumni networks are going to be full of important people as well.
Conclusion: Go to the best school for the thing you want to study. That's where the best companies in your career field are going to recruit from, and likely that is the school all the most prestigious people in your career field attended themselves.