Limbo as a third final resting place is mere theological speculation, and this speculation has already been condemned by Pope Pius VI. Limbo as a fringe of Heaven is also erroneous theological speculation.
However the Church has taught a limbo of Hell (for those who die in original sin):
Auctorem Fidei - Pope Pius VI
The Punishment of Those Who Die with Original Sin Only
26. The doctrine which rejects as a Pelagian fable, that place of the lower regions (which the faithful generally designate by the name of the limbo of children) in which the souls of those departing with the sole guilt of original sin are punished with the punishment of the condemned, exclusive of the punishment of fire, just as if, by this very fact, that these who remove the punishment of fire introduced that middle place and state free of guilt and of punishment between the kingdom of God and eternal damnation, such as that about which the Pelagians idly talk,—false, rash, injurious to Catholic schools.
The Council of Florence also taught this doctrine:
“Also, the souls of those who have incurred no stain of sin whatsoever after baptism, as well as souls who after incurring the stain of sin have been cleansed whether in their bodies or outside their bodies, as was stated above, are straightaway received into heaven and clearly behold the triune God as he is, yet one person more perfectly than another according to the difference of their merits. But the souls of those who depart this life in actual mortal sin, or in original sin alone, go down straightaway to hell to be punished, but with unequal pains.” (Florence, Session 6, July 6th, 1439).
The above teachings are perhaps not infallible, but they are teachings of at least the ordinary Magisterium.
The Church has taught a limbo of the fathers. For example, this doctrine, in at least some form, is necessarily implied by the following teaching:
On the Beatific Vision of God
Constitution issued by Pope Benedict XII in 1336
By this Constitution which is to remain in force for ever, we, with apostolic authority, define the following: According to the general disposition of God, the souls of all the saints who departed from this world before the passion of our Lord Jesus Christ and also of the holy apostles, martyrs, confessors, virgins and other faithful who died after receiving the holy baptism of Christ- provided they were not in need of any purification when they died, or will not be in need of any when they die in the future, or else, if they then needed or will need some purification, after they have been purified after death-and again the souls of children who have been reborn by the same baptism of Christ or will be when baptism is conferred on them, if they die before attaining the use of free will: all these souls, immediately (mox) after death and, in the case of those in need of purification, after the purification mentioned above, since the ascension of our Lord and Saviour Jesus Christ into heaven, already before they take up their bodies again and before the general judgment, have been, are and will be with Christ in heaven, in the heavenly kingdom and paradise, joined to the company of the holy angels. Since the passion and death of the Lord Jesus Christ, these souls have seen and see the divine essense with an intuitive vision and even face to face, without the mediation of any creature by way of object of vision; rather the divine essence immediately manifests itself to them, plainly, clearly and openly, and in this vision they enjoy the divine essence . Moreover, by this vision and enjoyment the souls of those who have already died are truly blessed and have eternal life and rest. Also the souls of those who will die in the future will see the same divine essence and will enjoy it before the general judgment.
It cannot be said, generally and without qualification, that the Church has never taught limbo in any form.