There are two types of stability when you talk about bullets, gyroscopic and dynamic. Gyroscopic stability is what most people talk and know about and is the result of the bullet being spun. The spin overcomes the overturning forces exerted on the projectile nose. Those forces are highest at the muzzle. As the projectile slows down, the overturning forces on the bullet nose decrease, but the spin degrades at a much slower rate than velocity. The result is that as long as a bullet is stable at the muzzle, gyroscopic stability will increase as the bullet goes down range.
Dynamic stability is more complex and has to do with how well a bullet negotiates the transonic area of its flight. The classic example of this is the .30 cal 168gr. bullet fired from an M-14. That combination just doesn't shoot accurately at 1000 yards because the 168gr bullet is dynamically unstable.
When you say stabilize, do you mean spin?
With match grade projectiles I don't think that you can spin them too much within reason. With ball bullets where the jackets are of poorer quality then the answer is yes. In the UK where they formerly used issue 7.62mm ball ammo for long range (800, 900 & 1000 yard) target shooting, they typically used 1/13 or 1/14 twist barrels. The reason was the jackets had problems with uniformity. One side of the jacket was a little thinner or thicker. When you spin these up out of a rifle, you get a wobbling because of the slight difference in mass. This is a picture for illustrating uniform case necks, but the same is true for bullet jackets
They found that spinning the bullets at the minimum required for stability resulted in greater accuracy. However match grade bullets by their nature have very uniform jackets. As an example of this, John Whidden, one of the best long range Palma shooters in the US uses a 1/10 twist barrel for shooting 155gr Palma bullets. Conventional wisdom was that he was spinning them too much, but it has not seemed to hinder his ability to win matches.
B