1. Do the language proficiency exams for non-native speakers correspond accurately to the CEFR?
2. Are there language proficiency exams for native speakers of your mother tongue or language that you are learning?
I'll give some examples for Chinese:
1. Hanban, the organisation that offers the HSK, a Chinese proficiency exam for non-natives, claims that the highest level (HSK 6) of the exam correponds to C2, but the associations of Chinese teachers in some European countries place the highest level much lower in the CEFR from as low as B1.2 to max. C1.1: https://en.m.wikipedia.org/wiki/Hanyu_Shuiping_Kaoshi. The German association of Chinese language teachers sees the HSK 6 equivalent to level B2. So, with a HSK 6 certificate I could theoretically claim being highly advanced but in reality I'd be just upper intermediate.
2. There are a written and an oral exam for Chinese native speakers in China.
ZHC for written Chinese: https://en.m.wikipedia.org/wiki/ZHC
Putonghua Proficiency Test for spoken Standard Chinese: https://en.m.wikipedia.org/wiki/Putonghua_Proficiency_Test.
If I'd take classes for accent reduction in Chinese, I'd choose a teacher who passed the highest level of this second exam.
In any case, Chinese native speakers have the possibility to prove their native proficiency with those exams.