The codon used for translating DNA/RNA code to protein is well established. It's a three-base degenerate code, meaning that there are several three-base DNA sequences representing a given amino acid [1]. This code is very well understood. If your DNA/RNA sequence has any of the three bases combos for alanine, your protein gets an alanine in that position. It follows from this that different DNA sequences can code for exactly the same amino acid sequence in a protein. However, proteins with the same amino acid sequence are chemically and biologicalally identical (ignoring things like post-translational modification).
A few years ago, I read a paper [2] where the group hypothesized that in a specific case, a rare codon for for an amino acid in a specific protein caused the cellular machinery to stall at that position. They suggested that in the intervening time, the protein misfolded into a different 3D shape. The resulting protein therefore had different chemical properties despite having identical amino acid sequence. Basically shredding what is often known as the "central dogmal of molecular biology".
Now, this specific example probably needs to be confirmed, and might not be very frequent. But it makes total sense when you understand how all the pieces work. However this explanation would be very low on most biologist's lists of reasons why a certain protein isn't functioning properly. In fact, when people do genetic analysis looking for diseases, they routinely throw out all synonymous changes before doing the stats. It makes you wonder how often we miss this when looking for disease genes.
My larger point is that lots of biological science is a collection of edge cases. We know so little about the systems we're studying and have such crude tools to investigate them, that we get blindsided by things sitting in plain sight all the time.
[1] http://en.wikipedia.org/wiki/Genetic_code [2] http://www.sciencemag.org/content/315/5811/525.abstract