A friend of mine did an engineering postgrad, and his doctoral thesis, based on materials, was a two part examination of the facts that, according to known laws of physics and engineering,  a) the Clifton Suspension Bridge should have collapsed the minute the first coaches began to cross it, and b) the drive shaft of the Austin Allegro should have sheared immediately upon reaching 30mph. Now, whilst I'm no physicist or engineer, and couldn't follow his arguments, the fat that he got his doctorate suggests that there are definitely some things which current scientific understanding cannot explain!

(I know there are well-known mysteries like the bumble bee and all, but I like these two better, because they have more personal resonance!)

JonTheCelt