No we would not advise putting Chilly's Bottles in the fridge. The reason is that this will have no effect on the Chilly's Bottle due to the double wall vacuum insulation. This technology means the Chilly's Bottle is not affected by the outside temperature so the fridge will not be able to cool the water inside the Chilly's Bottle.
For best results for cold water, we recommend using very cold water from the fridge or tap and filling your Chilly's Bottle with ice cubes.
We sell a fridge bottle so you can ensure you always have cold water: https://shop.chillysbottles.com/collections/accessories/products/chillys-fridge-bottle (UK only)