Amanda is correct, the issue is having a group of the same species, and 6 is generally considered the minimum for most characins, though some need more as I'll explain in what follows.
The reason is that these are shoaling fish, not to be confused with schooling fish. Technically, the two terms are quite different. Freshwater fish shoal (at least those species that are "shoaling" do), which means they occur together in large numbers in the wild. They use this shoaling for protection ("safety in numbers" idea), and many have some level of social hierarchy within the group. When they are on their own, or in 2 or 3, they feel vulnerable and in danger, and that causes stress. Stress affects the immune system, so the fish is more likely to come down with various ailments which would not occur if the group was larger. Those that have more developed social structures also can get more aggressive when the expected group is not present. Recent scientific study for the first time has proven that shoaling fish not kept in groups will show considerably more aggressive behaviour to their own and other species in the tank. And that heightens the stress for the other fish, which again means more health issues. I expect you get the picture.
Most species of characins (tetra, hatchetfish, pencilfish) and cyprinid (rasbora, barbs, danios, loaches) are shoaling fish by nature. Some species have more of a social framework within the group, and need even more numbers. Serpae Tetra
and Tiger Barbs are two that come to mind; these fish are generally recommended to be in groups of no less than 8 and preferably more than that. Part of the reason is to reduce aggression on subordinate members of the group. The dominant fish are less likely to severely weaken the subordinate fish if there are more of them to spread it around. Species like Cardinal Tetra
and rummynose tetra do not have this heightened instinct of dominance and thus they are fine in a group of 6 although where space is available more is still better. Rummynose should always be in a group of 12 or more.
You may not see anything different between 3 or 6 bloodfins; but I can assure you the fish will be much more "relaxed" with 6 or more, and less stress means better health.
"Schooling" behaviour generally means fish that stay together in a group, swimming as a unit. Very few freshwater fish do this, unlike marine reef fish. However, some freshwater fish do swim generally together as a group, rasbora are particularly prone to this, as are rummynose tetra. It is interesting to observe the fish's natural behaviours when they have space in the tank. I have a 115g aquarium which is 5 feet long, and there are several species of characin. The two rummynose species are almost always together, the Cardinal Tetra
are in the same general area as a group, the Pristella, rosy and bentosi tetras tend to stay together (all three species) sort of mixed but still within sight of their own species, and these three tend to occupy one half of the aquarium. None of these ever "attacks" any of the others, but they have their own areas and they tend to remain there, together. We would say they are "happy" and that means healthier.
So in your case, another 3 bloodfins definitely. Then, if you have sufficient space for another shoaling tetra in a group of 6+, there are several that will be fine with bloodfins. Have a look at our fish profile section, second tab from the left in the blue bar at the top; under "characins" there are I would say all of the commonly-seen species of tetra included, and a few that are less common.
Hope that explains it a bit.