Will these numerous B's hurt me greatly?

As a Junior with a not-so-fantastic GPA, I wont be looking at schools ike HYP, UPenn, Northwestern, etc. However, I will be looking toward schools in the top 20-40 schools, like UC Berkeley, Case Western, Boston College.

Freshman Year:

Honors English: A-
Honors Algebra 2: A
Spanish 2: A-
Honors Physics: B+
Advanced Chamber Ensemble: A
Honors World History: A

Starts to go downhill a little bit… in its defense though, my mom got really ill, got depressed/demotivated, had family fights, and somewhat screwed the year up.

Sophomore Year:
Honors English 2: A-
Honors Precalc: B+
Spanish 3: B+
Honors Chem: B+
Advanced Chamber Ensemble: A
AP Euro: A (5 on the actual test)
Sports Training (which probably wouldnt count XD): A

Junior Year (this is how it will end if I keep my grades up):
AP English: A-
AP Calc AB: B+
Spanish 4: A
AP Chem: B+
Journalism: A
AP US: A-
Bio Honors: B or B+

Overall, my UW GPA should be a 3.69. D:

There’s a lot of horror stories here about people with 4.0 gpas w/ 2300 SATs getting rejected from UCLA, USC, etc, along with 10+ great ECs. Having said that, will those numerous B+s greatly reduce my chances?

Depends on how your school is viewed. I had a 3.52 UW with a slight downward trend at an elite prep school and was able to get into a top 25 national LAC (Richmond) and waitlists at a top 20 national university (Vandy) and a top 10 national LAC (Davidson). I did get rejected by two top 10 national universities (Duke ED and Penn-Wharton). I also got a half scholarship and honors at a top 50 national university (Miami) and the same to one just outside the top 50 (Tulane). What helped me was a very high SAT score (2310) and a 4.46 Weighted GPA due to taking all honors (we don’t have AP).

You can definitely get into a school ranked somewhere between 20-40. Remember, your transcript is only one fraction of the entire admissions process. They take into account your SATs, extracurriculars, essays, race, community service, etc. I can’t speak about your chances of getting into Berkeley without the other information, but I can say you have a pretty good shot.