If X is defined over a field of characteristic two, then there is a natural line bundle like in your question that you can take. Namely, in terms of divisors, the divisor of an exact differential dx is of the form 2D. Use the line bundle corresponding to D and consider first etale covers of X with degree a power of 2. If X is ordinary your H^0’s will be trivial and if X is not ordinary they will grow linearly with the degree. The proof is not difficult but is too long for a blog comment. Now, if X is ordinary but has a non-ordinary etale cover, then you get unbounded H^0 by taking covers of that cover. Does that work for all X? Maybe, I don’t know. Unfortunately, this does not help in characteristic zero even if X has good reduction at 2, as the inequalities for H^0 go the wrong way.
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