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Look at the strong converse to Shannon's theorem: for rates above the channel capacity, if $n$ bits are to be transmitted, the probability of error is exponentially close to 1, so $1-e^{c n}$ for some constant $c$ depending on the channel. Also, look at rate distortion theory. This gives a formula for the highest rate at which you can transmit if you ...
This is actually problem 5.12 in Cover and Thomas's information theory textbook; show that the probability distribution ${1/12,1/4,1/3,1/3}$ gives a counterexample. And if you want a really nice counterexample, consider the many non-isomorphic Huffman trees you can make when you have probabilities proportional to $$1,1,1,2,3,5,8,13,21,34$$ (the Fibonacci ...