Portfolio (of real work) > degree
Experience + portfolio > degree
Contacts + experience + portfolio > degree
Versatility + experience + contacts + portfolio > degree
I can't disagree with Blue Velvet's quote above in abstract... but there are some complexities you should be aware of...
The different degrees - associates, bachelor of arts, and the bachelor of fine arts - are essentially defined by the number of hours you spend in studio, working under faculty supervision. Each degree, in order, requires more time in studio (and conversely less time on other courses) working on your art and design work.
In a great many percent of cases, student portfolios directly reflect the amount of time students have had in studio. This means that the more time you spend in studio, the stronger your portfolio will be.
I look at a lot of student portfolios, and while you do get "superstars" at every education level, on average the portfolios reflect the students education level...
for example, a design student with a B.A. may have taken one semester, or perhaps a year, of typography. A B.F.A. student usually has between three and four semesters of typographic study. Twice as much time working in a discipline is going to be visible in the student portfolio.
Expertise comes with time... and the different degrees are there to allocate time differently within your education.
In the United States, the B.F.A is designated the "professional degree" from an accreditation standpoint... it is a degree designed to prepare students for industry.
So, a designer is always judged by the quality and sensitivity of their portfolio... and the professional world does separate the good from the bad... but at the student level, as you are hustling for those first couple of jobs, your education level does come into play because it is reflected in the things you have made.
I will say that I see a lot of students with associates degrees returning back to school after a few years to get a more advanced degree. That may not be true in all areas, but in a competitive area like mine -- we have two freestanding colleges of art and design in my city graduating over 80 Graphic design BFAs a year -- the students find it hard to compete against the more advanced students to get jobs and also find it harder to advance up the career ladder once they have a job because they have not had the same level of training in formal and conceptual skills.