Where does the Bible tell us that the church should, or someday would be divided into competing and disagreeing denominations?

Not 'should', only 'would':



1 Corinthians 3:6 I have planted, Apollos watered; but God gave the increase.
7 So then neither is he that planteth any thing, neither he that watereth; but God that giveth the increase.
8 Now he that planteth and he that watereth are one: and every man shall receive his own reward according to his own labour.
9 For we are labourers together with God: ye are God's husbandry, ye are God's building.

10 According to the grace of God which is given unto me, as a wise masterbuilder, I have laid the foundation, and another buildeth thereon. But let every man take heed how he buildeth thereupon.
11 For other foundation can no man lay than that is laid, which is Jesus Christ.
12 Now if any man build upon this foundation gold, silver, precious stones, wood, hay, stubble;
13 Every man's work shall be made manifest: for the day shall declare it, because it shall be revealed by fire; and the fire shall try every man's work of what sort it is.
14 If any man's work abide which he hath built thereupon, he shall receive a reward.
15 If any man's work shall be burned, he shall suffer loss: but he himself shall be saved; yet so as by fire.

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The times of this ignorance God winked at; but now commandeth all men every where to repent:

Behold, I stand at the door, and knock: if any man hear my voice, and open the door, I will come in to him, and will sup with him, and he with me.-kjv