This question requires careful considerations for a safe answer.
Water can be filtered safely from many natural water sources, such as streams, river and lakes. That being said, there too are many water sources that are simply too contaminated to be drunk, and wouldn't be safe to use, regardless of how many times the water may be filtered.
Signs Of Bad Water
There are some telltale signs that a water source should not be used, as these water sources may not be safe even after boiling. You can often tell by the smell of the water source whether or not it can be considered as a drinking source. This is an important reason, as really bad sources may even have corpses of animals either floating within, or close to the edge. These water sources should be avoided at all costs, as there is no reliable way to know in the field whether or not the water can be made safe to drink by either filtration or boiling.
It's important to exercise caution when selecting your water source, and good judgement on choosing a water source can result in safe drinking water, without the need for further purification.
As of late, there are only a few water sources in the UK from which nobody could ever drink. Rivers by and large are fairly safe... especially fish-bearing waters. Likewise, a lake containing fish is likely to be cleanable.
For top tips to deal with contaminated water outdoors, read Tips for Dealing With Contaminated Water In The Backcountry.
The Best Option
When we filter water with our AquaHike Straw, we choose to use it on moving natural water sources, with no additional purification (no chemicals or no boiling). We do this with mountain streams and rivers mainly. Similar for filtering collected rainwater from puddles.
When drinking from ponds and lakes, if often depending on the smell, clarity of the water, presence of life and absence of corpses. If it passes all the tests, filtering the water before drinking can be all that's needed. If you have even the slightest doubt in the water quality, filter first, then boil the water for at least 60 seconds, at a rolling boil. This will kill any harmful viruses that the AquaHike Straw couldn't remove.
We love using AquaHike Straw every week, and we've never reported being sick as a consequence of that.
For a detailed look at filtration vs purification in the wild, read Filtration vs Purification.
So just remember: if in doubt, boil. If it looks or smells like death, don't even consider it.
To learn more ways to access clean water in the wild, read 7 Ways to Access Clean Water in the Wilderness.