This applies to the case when the table rates shipping method was set up to be in “weight vs. destination mode”. In addition to this, another symptom is: the table rate shipping costs shown are always zero. Even though non-zero shipping costs are configured for zero weight in the table rates CSV file. This was experienced in Magento CE 126.96.36.199.
This problem is also found in the WebShopApps Premium MatrixRate extension version 4.5.1 on Magento CE 188.8.131.52, so seems to be a problem in the Magento core.
The reason was, in this case, that the frontend user used for ordering had a 100% shipping discount set up for his customer account. This normally leads to zero-ing the shipping costs of all selectable options, but also seems to make the shipping weight to appear as zero. Which should be considered a bug, as there is no reason for that.