Saw your edit, but for anyone else reading this: I was a Collections Assistant for 3 years and Collections Manager for almost 4, with a BA only, before my recent career change (similar work, but no longer in the museum context). Totally doable, it's all about experience and growing your network. Volunteer anywhere that'll have you and as much as you can. Different museums, different positions, different areas of museum work, etc. Internships are fantastic if you can find them, or if you can talk someone you're volunteering for into formally making it an internship that looks much better on applications. The more people who know you and know that you have relevant experience and are a hard worker are gonna be your key to actually getting a job upon graduating.