I don't think true contradictions are actually tolerated in Biblical Hermeneutics.
I think what we see is that God often uses teachings which appear to contradict (at first glance) to clarify things that we otherwise might have glossed over.
In the following passage, Jesus uses two contradictory statements in the same sentence:
‘I know your tribulation and your poverty (but you are rich)' -Revelation 2:9
This 'contradiction' causes us to consider what Jesus meant by the words poverty and rich in this passage.
We see the same sort of thing in this passage concerning a person's life:
For whoever wishes to save his life will lose it, but whoever loses his life for My sake, he is the one who will save it. -Luke 9:24
Again, this glaring "contradiction" causes us to pay closer attention to the phrase and ask "what does He mean by that?"
The Law of Non-Contradiction basically says that something can not be both true and not true at the same time in the same way. I do not believe Scripture violates this Law.
In the first example, we could use other Scriptures to show that the terms poverty and wealth can refer to either the material realm or the spiritual realm. A case could also be made that one can be rich technically speaking, but can live in poverty for the sake of giving more. There are other options as well.
In the second example, we could use other Scriptures to show that we must "lose our life" in the sense of "laying down our life" and submitting it to the will of God, "dying to our old self", "crucifying our passions", and so forth. By doing so, our life is "saved" in a different sense of the word 'life' - we are given "new life" as a "reborn creature", and begin to live for Him, with the final reward of "eternal life".
In the same way, the doctrine of the Trinity does not violate the Law of Non-Contradiction, because the doctrine simply states that God is one in one sense and three in another sense.
Hope that helps.