Harvard, Yale, MIT, and Stanford. The safe school (I hope I don't have to go there) is UCB.
My main concern is that would an apartment be to small for a German Shepard?
As a proud alumna of one of those schools, may I humbly suggest researching how to spell the breed name?
My DH, a Caltech alumnus, just observed that he would expect students at MIT to write about "Shepards." Just sayin'....
Sorry, we couldn't help it--the MIT/Caltech rivalry is deep--esp. in the Physics Dept.
As to your question: no, you won't have time for a dog. At any of those schools, if you fail to immerse yourself in all that they have to offer in order to spend a lot of time goofing off with a dog, you'll have wasted an opportunity. If you do things properly, you'll soon be very, very busy with extracurricular activities, brilliant friends who inspire you, independent research, and advanced seminars.
Moreover, living in an apt. in Cambridge or New Haven with a dog in winter would be positively dreadful at your age. Living with a dog in Palo Alto or Berkeley would be marginally better because of the weather, but you still won't have time for it. All four of those cities also have very tight housing markets, so finding a rental with a dog would be difficult -- unless you don't mind living in a sketchy area, far from campus. The best social life will be on campus, in the dorms, your Freshman year. Don't underestimate how worth knowing your fellow students may be. One of the most important aspects attending an elite institution is sharing ideas with the great minds all around you--and that as often as not happens over a beer on a Sunday afternoon as in the library during a study session. You'll miss out on that if you have to spend the day training or exercising the dog because it's been cooped up all week.