First of all, I'm a linguist asking a VERY technical question, so I'll do my best to make myself understood. I'd prefer if commenters made it clear what their level of Japanese is (native, advanced, etc.).
English and Japanese are said to differ in that the former has tense harmony, whereas the latter doesn't. Having tense harmony means that if a main verb and its subordinate are both in the past tense, then the whole clause must be interpreted as referring to two simultaneous past events. For instance, (1) means that at some time t in the past Taro said that Hanako was in Seattle at t (while he was in Tokyo speaking).
(1) Taro said that Hanako was in Seattle.
Japanese works differently. The literal, word-by-word translation of (1) into Japanese means that at some time t in the past Taro said that Hanako was in Seattle at t', for some time t' < t (in particular, Hanako could've already left Seattle when Taro made his statement):
(2) 太郎は、花子がシアトルにいたと言った (back-shifted interpretation)
In order to get a proper translation of (1) you need a present tense verb in the subordinate clause:
(3) 太郎は、花子がシアトルにいると言った (simultaneous interpretation)
This shows that Japanese doesn't have tense harmony: when the main verb and its subordinate are both in the past, you don't get "harmony" or "simultaneity"; you get a back-shifted interpretation in which the event referred to by the subordinate verb is a past event relative to the event referred to by the main verb.
However, it is claimed that for some Japanese speakers the "simultaneous reading" is admissible when the main verb is a factive verb [What is a factive verb? A factive verb presupposes that its subordinate clause is true: know is factive because "she knows that it's raining" presupposes that it's actually raining; on the other hand, believe is not factive because "she believes that it's raining" doesn't presuppose that it's raining... she could be dead wrong!].
Does this accord with your intuition? Do you interpret (4) to mean that Taro knew at t that he had cancer at t, or do you get a back-shifted reading? Are there any more examples like this one? (I know very little of Japanese, but I'm told other factive verbs are 後悔する, 忘れる, etc.).