While I understand the rounding issues, I must continute to question the precept.
In my physics class in College, the professor told us on all tests that he only wanted 1 significant digit. Pi/3 = 1. This was an attempt on his part to break people of their calculator habit (he didn't allow them into tests). Practically speaking though, 3.1415927/3 = 1.04719756. Let's assume that this actually represents a real unit like Meters. You send the part to get machined. Where do the digits become irrelevant to the machinist? There may be some applications where you need the part manufactured down to the millimeter. Down to the micrometer though, is unlikely, since expansion and contraction at any temperature for just about any substance will exceed that tolerance. (I can't completely support this, there may be some new materials that that don't contract and expand much).
I can't say that making calculations as precise as possible is bad. Isolating possible points of failure is never bad. From my experience though, I think they are focusing on the wrong point of failure. There are very practical reasons for the lessons we receive in Significant digits. They apply to gross computational methods as well as hand calcs.
As to Global Circulation Models, I can only say that anyone attempting to model a system without knowing the boundary conditions should not expect much in the way of accuracy or precision. To stand by such models and say they are accurate is downright fraudulent. The inaccuracies of such models are not due to significant digits, they are due to an incomplete understanding of the system they are modeling. Once again it is not bad to try and model the system to learn from the process and try and model it better. It is only bad to misrepresent the accuracy of the models.
100 Bit mantissas will do nothing to make the accuracy of such models increase. Decreasing the size of the node, on the otherhand might. The only reason that a larger mantissa seems useful is because the last digits 'might' represent a molecule of air. This is faulty though because the size of the node (I believe they are about 100Km on a side and 1 km deep), because the trailing digits are irrelevant to the magnitude of the answer.
If you really want to increase the accuracy of such models, you will have to model down to the molecular level, but that would require extraordinary amounts of energy. Modeling my office would require more computing power than is currently being used on GCM's now.