The most grammatically versatile of all English words is naturally prone to a wide variety of abuses. Here we focus on those ambiguities in the usage of that that arise as a direct result of its versatility. Our first observation is that while the same word may legitimately serve as relative pronoun, demonstrative adjective or adverb, or subordinating conjunction, it should always be clear which one of these roles is being assumed in any given context. This principle is frequently violated, as in the following example:

This is the date that Pearl Harbor was attacked.
This might be characterized as a dihedral usage. When the pivotal that faces to the left and is considered in the context of This is the date that ... , it gives the appearance of a relative pronoun, as in
This is the date that will live in infamy.
But when it turns to the right, it becomes a part of the clause that Pearl Harbor was attacked and is thereby transformed into a subordinating conjunction, as in
We were shocked to learn that Pearl Harbor was attacked.
In such situations, it is probably best to avoid the use of that altogether; in the example above, it may effectively be replaced with on which.

As a relative pronoun, being equally proper in both subjective and objective cases, that is disposed to another sort of duplicity, which does not threaten who, whom, and other case-specific pronouns. Thus, since whom cannot legitimately assume the role of who, the sentence below offers no temptation to economize by deleting the latter:

He was a friend whom I trusted implicitly and who never disappointed me.
On the other hand, it is perhaps less obvious that the second that is indispensable in the following:
This is a problem that warrants my attention and that I may be able to solve.
Its removal, however, leaves the sentence not merely less readable but strictly ungrammatical, requiring the remaining that to serve in both cases at once:
This is a problem that warrants my attention and I may be able to solve.

It may also be appropriate here to make note of a common error of an opposite sort, whereby two conjunctive instances of that are used where one is sufficient:

We know that by the very nature of the creative process that we are one with the Originating Spirit... *
Of course, this blunder is more difficult to spot when the two thats are separated by a substantial section of text. The following example is further complicated by two pronominal instances, which are apt to numb the reader to the redundancy of the final conjunction:
The worst possible outcome is that they get off, that somehow through all of the discussions that we've been having, and somehow through releasing evidence that we have, that the people who deserve to be punished are somehow released. *
One could hardly hope to improve on Fowler's absorbing account of this phenomenon, which he terms the "interim that":
It often happens to a writer to embark upon a substantival that-clause, to find that it is carrying him further than he reckoned, & to feel that the reader & he will be lost in a chartless sea unless they can get back to port & make a fresh start. His way of effecting this is to repeat his initial that. This relieves his own feeling of being lost; whether it helps the inattentive reader is doubtful; but it is not doubtful that it exasperates the attentive reader, who from the moment he saw that has been on the watch for the verb that it tells him to expect, & realizes suddenly, when another that appears, that his chart is incorrect.