Человек сдает экзамены, чтобы попасть в магистратуру. Это не то же самое, что магистерские экзамены, которые сдают, чтобы получить степень магистра.

Mariia, a "master admission exam" is any principal (=master) examination for entry (=admission) into any company or institution. However, in reference to the MasterS degree in higher education, it's a MasterS exam.

Алтынай, that would be how Borat would describe his school experience.

When we talk about taking exams, we use the personal possessive form: ...taking MY xxx exams.

Larisa: MY masters degree exams

Hi Uly, so Masters Exam(ination)s are those needed for the enrollement and the (final) exam(ination) for the Master's degree would be the last exam in the Master's programme, right?

Correct. The Masters exam gets you into the program, and then you take the Masters FINAL exam(s). https://www.google.com/search?biw=1374&bih=1267&q=%22masters+final+exam%22&oq=%22masters+final+exam%22&gs_l=psy-ab.3..0l4.706294.707446.0.708164.10.10.0.0.0.0.144.895.6j3.9.0....0...1.1.64.psy-ab..3.7.633...0i7i30k1.0u-kHY0cw-A

Ok, Uly, thanks.
One more question. Why do they talk about postgraduate admission exams here (far master's programmes as well)?: http://www.fulbright.org.uk/going-to-the-usa/postgraduate/educationusa-advice/applying/admissions-exams/which-exam-to-take

I imagine they're referring to exams for admission into postgraduate programs.