A few years ago the International Theological Commission published a
paper expressing the position that the theory of Limbo is not necessary and we can still hope for the full salvation of unbaptized babies.
It’s important to note that, first, Limbo was never formal Catholic doctrine and I know for a fact that some Catholics expressed disbelief in Limbo before this paper came out. Second, the International Theological Commission, though connected to the Vatican, is not (currently, or at the time this paper came out) an arm of the Magisterium. It has no teaching authority in the Church, just the ordinary scholarly authority of its members. And third, even the ITC paper acknowledges that Limbo “remains therefore a possible theological hypothesis.”
On the other hand presumably Pope Benedict XVI could have stopped the paper from being published and certainly from it being made available on the Vatican’s website, so the fact that he didn’t suggests he may agree with its conclusions, and even apart from his office he is a very well respected theologian.
Therefore, if a Catholic wants to continue believing in Limbo, he or she may do so, though it might be a good idea to give the ITC’s arguments consideration. Personally the theory that makes the most sense to me is that God miraculously gives babies a chance to choose or reject him immediately before their deaths and that choice determines their eternal destiny, but I’d wager fewer Catholics believe in that than Limbo. The most common two approaches I’ve seen from fellow Catholics (and I’ve done no polling and statistics, of course), especially since this paper has come out, is either to assume all babies go to heaven or simply express hope that they do and leave it at that.