She told me she was a student.
Можно ли сказать she said me she was a student?
Если нет, то почему?

Можно сказать told me, said to me или просто said:She told me she was a student.She said to me she was a student.She said (that) she was a student.

нельзя говорить She said me. Либо She said that she was a student или She said to me she was a studentЛибо She told me she was...Почему? потому что правило такое, после say буз предлога можно добавлять только прямое дополнение: say hello, say goodbye, say a word