There are two apparent puzzles connected with the two-body and three-body doubly charmed baryonic B decays. First, earlier calculations based on QCD sum rules or the diquark model predict B((B) over bar (0)->Xi(c)(+)(Lambda) over bar (-)(c))approximate to B((B) over bar (0)-> B-c(N) over bar), while experimentally the former has a rate 2 orders of magnitude larger than the latter. Second, a naive estimate of the branching ratio O(10(-9)) for the color-suppressed three-body decay (B) over bar ->Lambda(c)(+)(Lambda) over bar K--(c), which is highly suppressed by phase space, is too small by 5 to 6 orders of magnitude compared to the experiment. We show that the great suppression for the Lambda(c)(+)(Lambda) over bar K--(c) production can be alleviated provided that there exists a narrow hidden charm bound state with a mass near the Lambda(c)(Lambda) over bar (c) threshold. This new state that couples strongly to the charmed baryon pair can be searched for in B decays and in p (p) over bar collisions by studying the mass spectrum of (DD(*))-D-(*) or Lambda(c)<<(Lambda)over bar>(c). The doubly charmful decay B ->Xi(c)(Lambda) over bar (c) has a configuration more favorable than the singly charmful one such as (B) over bar (0)->Lambda(c)(p) over bar since no hard gluon is needed to produce the energetic Xi(c)(Lambda) over bar (c) pair in the former decay, while two hard gluons are needed for the latter process. Assuming that a soft q (q) over bar quark pair is produced through the sigma and pi meson exchanges in the configuration for (B) over bar ->Xi(c)(Lambda) over bar (c), it is found that its branching ratio is of order 10(-3), in agreement with the experiment.