Lie’s theorem indeed still holds in positive characteristic provided the dimension of the vector space is less than the characteristic. For reference see, for example, the remark before example $81$ in http://math.berkeley.edu/~reb/courses/261/11.pdf, It is indeed often the case that results being true in characteristic zero, remain true for $p$ larger than the dimension of the vector space (or larger than the Coxeter number for simple Lie algebras). On the other hand there may be many results from modular representation theory where we need to be careful with such a statement.