他有一些成绩，但也不应该骄傲。
My translation : He got some good achievements/grades, but he shouldn't be conceited.
I've got the meaning "but also" translating directly 但也, but seems this translation doesn't fit in this case.
Nextly thought that adding 连 at the beginning of the sentence namely composing following.
连他有一些成绩，但也不应该骄傲。 : Even he got some achievements/grades, he shouldn't be conceited
I suspect this is also incorrect.
I want your wisdom.
也 here means "still"
他有一些成绩(he got some success)，但也不应该骄傲 (but still shouldn't be cocky)
Another way to look at 也(even)"
有一些成绩(也) = (even) got some success
不应该骄傲 = shouldn't be cocky
有一些成绩(也)不应该骄傲 = shouldn't be cocky (even) got some success
More examples:
打死我[也]不去
Will not go [even] if someone beat me to death
or
You can beat me to death and I [still] won't go
一年薪水[也]不夠買車
Not enough to buy a car [even] with a whole year's salary
or
A whole year's salary [still] isn't enough to buy a car
