#Java, 407 398 351 308 bytes
Java, 407 398 351 308 bytes
Golfed it with help from @Geobits and @SamYonnou
Spread the word: Verbose == Java
import java.math.*;class A{public static void main(String[]w){BigInteger a=BigInteger.ZERO,b=a.flipBit(0),c,z=a,t=a.flipBit(1),h=t.flipBit(0),s=t.flipBit(2);for(int i=0;i<100;i++){System.out.println(b.mod(s).equals(z)?"FiboNacci":b.mod(t).equals(z)?"Fibo":b.mod(h).equals(z)?"Nacci":b);c=a;a=b;b=c.add(b);}}}
Ungolfed version:
import java.math.*;
class A
{
public static void main(String[]w)
{
BigInteger a=BigInteger.ZERO,b=a.flipBit(0),c,z=a,t=a.flipBit(1),h=t.flipBit(0),s=t.flipBit(2);
for(int i=1;i<=100;i++) {
System.out.println(b.mod(s).equals(z)?"FiboNacci":b.mod(t).equals(z)?"Fibo":b.mod(h).equals(z)?"Nacci":b);
c=a;a=b;b=c.add(b);
}
}
}