I graduated from Brown University in 1997, and I majored in Math, not Computer Science.
Every so often a "Is an Ivy worth it / what's my degree situation going to matter?" conversation comes up online, here's my take on your question.
In short, in my experience, it matters, in the following ways:
a) Social Networks. The social networks you build at an Ivy / MIT / Stanford / U of C will be more nationally useful. In any given city outside their zone, they will not matter as much as the local "good" University, though. For example, right now, I live in Seattle -- going to the University of Washington would have opened many more doors than my Brown degree does here. Harvard, Yale, Princeton, and to some extent Stanford transcend this dynamic, and are usually more valuable than even a pretty good local school; this is because the alums in town will tend to be significantly more influential.
b) Peers. IF (and this is a big IF for many who wonder about going to a given 'great' university that's not accepted them), you would have been an intellectual peer with people at your dream institution, you will have a correspondingly tougher time finding peers at the not-top-tier University. I have heard, many times, statements from second and third tier University attendees that their experience contradicts this; people are challenged, smart, etc. etc. at their local University. Out of curiosity for a while, I would interview transfers from other schools to Brown / MIT / Harvard, and 100% of the time they indicated that the other school was significantly less challenging, and offered far fewer peers.
Now, a counterpoint -- for a while, I was a member of the Young Entrepreneur's Network. Simplified member qualification: You had to own a majority of a business with more than $1mm in revenue to join. I met roughly one hundred fellow CEOs in this group for the couple of years I was in it; only one had an Ivy league education. I met no 'name' MBAs in this group.
The next step up, Young President's Organization has, I'm told, a few more MBAs, but I didn't qualify, so I can't confirm this.
To kick ass at business, you DON'T NEED TO GO TO A GREAT SCHOOL, in fact, in general, awesome success at real-world business ownership is contraindicated by an Ivy / top-tier education.
Of course, here on HN, we know that success fundraising an angel round in Silicon Valley is highly correlated with having a great technical degree, but it's even more highly correlated with executing in an awesome way.
If you want to do a great job starting businesses, learn to execute, and go crazy, don't worry about the rest.
On the other hand, if you want the network, intellectual challenge, and peer group and are sure you'll be unhappy anywhere else, there's a simple solution:
a) Get all A's at your current university
b) Start applying to your chosen university; reach out to professors doing interesting research, and work the admissions group. "I'm having trouble finding researchers at my University who can give me enough interesting work in X area, Professor so-and-so and I have corresponded, and I'd like to transfer." This speaks so much more than high school grades/SATs... Believe me. A recommendation from a professor that you're 'topping out' with would also work nicely. They'll want to help, in the best case.
The slightly less rock-star version is: crush your undergraduate, crush your test scores, do good senior year research, find some good grad programs, crush your GREs, and go to the grad school you want to go to. That's where you're going to spend a long time if you're serious about academia anyway.
On the other hand, if you can't get those kind of results at University of Newfoundland right now, I think you should relax -- you would not do well at MIT. I promise you. Put it out of your head, and go kick ass at business; it's significantly easier than excelling in academia at MIT.