The question is too vague for anyone to answer. The first inadequacy is that you don’t describe what you mean by filtered water. There is no specification for filtered water. What is it filtered with? A cartridge filter? depth filter? Sand? Multimedia? Green Sand? Carbon? Nano-filter? Ceramic ultra-filter. The list goes on and on.
As one helper pointed out, boiling will kill micro-organisms, but it won’t remove sediment and total dissolved solids (sodium, calcium, iron, chromium, etc). When you boil water, the dissolved solids (TDS) stay behind while the water escapes as steam. This means the concentration of metals in boiled water will be higher than the original water chemistry. Less water, same TDS = less pure water.
Also remember, boiled water will have no chlorine in it and as such is not potable water. This means that it can very well grow new bacteria and other organisms as soon as it cools. The message here is, don’t store boiled water longer than 24 hours, and refrigerate it as so as you are done boiling it.
If you have no choice but to boil water, then be mindful of the shortcomings.
In a nutshell, any time you start asking questions like this, especially if it is for guidance about what to do should your water supply ever be jeopardized; here is the best answer for you. Have a good supply of commercially packaged, unopened drinking water on hand.
Any time you get in to manipulating your drinking water, you bring on more health risks than you can imagine. If you are not a trained water technologist, don’t mess around with your water.
If you would like to know more about boiling water, here is part 1 of a multi-part series that might help.