<p>I don't think there would be a huge problem with this, depending on what teacher you end up with. If you do any research at all you will soon discover that the common wisdom for music people is that is not so much where you go to school but who you study with. And unless you are incredibly gifted, you usually cannot get in with the best teachers if you are not a performance major. It also depends on the students goals and what they hope to accomplish. That being said, there are number of professional musicians that I know of that did not major in music, even at the BA level. But the folks I know were already exceptionally accomplished before they started college and were able to continue to study with top teachers during their undergraduate years. </p>
<p>Ultimately, the pursuit of music as a career is extremely time consuming. My son, who is in a B. Mus program practices 3-4 hours each day, has 2 hour orchestra rehearsals 3 days a week, and rehearses with his chamber groups several times each week. On average he plays his instrument 6-7 hours a day. This is necessary for most people who aspire to be good enough to get into a good grad program. It is people like my son that students who pursue a BA will be competing with for grad school. That is not to say that the program you describe would not be successful for a dedicated student. But that student needs to realize that in an program that offers both B.M and B.A programs, the B.M candidates will almost always be given preference for teacher and performing opporunities.</p>