I ask this question based on the fact that tensile strength tests show that putting a double fisherman's knot in kernmantle rope makes it more likely for the rope to snap than a rope tied with a single fisherman's.
Knots weaken kernmantle rope, a lot in fact. Polymeric fibres are extremely strong along their longitudinal axis, however they have low flexural strength, meaning they are not strong along their horizontal axis, which is why ropes lose significant amounts of strength when tied. Knots create a stress concentration on a rope's horizontal axis, crushing the rope's core and causing it to fail under less force or weight than an unknotted rope. The tying of a double fisherman's knot will cause a measurable reduction (20-35%) in the tensile strength of a rope. Moreso than a single fisherman's knot (15-20%). The only real advantage that a double fisherman's has over a single is that it is less likely to slip, especially on newer synthetic fibre ropes, which is why it has mostly replaced the single in mountaineering.
It's fair to note that a double fisherman's knot still reduces the tensile strength less than most other knots, and that a knot isn't really useful if it slips, but with certain types of rope it's unlikely that even a single fisherman's will slip, whereas on fancy lightweight ropes–like aramid ropes–a triple fisherman's is necessary to prevent your rope from slipping.
If the rope or accessory cord that you are using will hold a single fisherman's knot without slipping, would it be better to use a single fisherman's knot (leaving really long tails) and take advantage of the higher tensile strength the knot provides over a double?