#Ruby, 71 66 bytes a=b=1;100.times{puts [b,f='Fibo',n='Nacci',f,b,f+n][~b%6];a=b+b=a}

###ungolfed:

a = b = 1 #starting values
100.times{
  # create an array, and selects a value depending on the current number
  puts([b, 'Fibo', 'Nacci', 'Fibo', b, 'FiboNacci'][~b%6])
  a=b+b=a # magic
}

Ruby, 71 66 bytes

a=b=1;100.times{puts [b,f='Fibo',n='Nacci',f,b,f+n][~b%6];a=b+b=a}

ungolfed:

a = b = 1 #starting values
100.times{
  # create an array, and selects a value depending on the current number
  puts([b, 'Fibo', 'Nacci', 'Fibo', b, 'FiboNacci'][~b%6])
  a=b+b=a # magic
}

#Ruby, 71 66 bytes a=b=1;100.times{puts [b,f='Fibo',n='Nacci',f,b,f+n][~b%6];a=b+b=a}

#Ruby, 71 66 bytes a=b=1;100.times{puts [b,f='Fibo',n='Nacci',f,b,f+n][~b%6];a=b+b=a}

###ungolfed:

a = b = 1 #starting values
100.times{
  # create an array, and selects a value depending on the current number
  puts([b, 'Fibo', 'Nacci', 'Fibo', b, 'FiboNacci'][~b%6])
  a=b+b=a # magic
}

#Ruby, 71 bytes f,n=%w[Fibo Nacci];a=b=1;100.times{puts [f+n,b,f,n,f,b][b%6];a=b+(b=a)}

#Ruby, 71 66 bytes a=b=1;100.times{puts [b,f='Fibo',n='Nacci',f,b,f+n][~b%6];a=b+b=a}