Yes, definitely, that general rule in English grammar applies to both single statements and multiple statements. We must clearly distinguish, though, between a “statement” and a “sentence.”
By definition, a statement is a single declaration, remark, or assertion that could be simply a single word of warning like, say, “Fire!”, or a long speech or perhaps a press release consisting of so many sentences or paragraphs. Obviously, a statement could invoke or involve several events or actions at different times, so to put those events or actions in context, the sentences describing them would have to use the appropriate tense for each of them.
In contrast, a sentence by definition is “a word, clause, or phrase or a group of clauses or phrases forming a syntactic unit which expresses an assertion, a question, a command, a wish, an exclamation, or the performance of an action…” (This is from the definition by the Merriam-Webster’s 11th Collegiate Dictionary.) Within a sentence, there are specific grammatical rules for verbs that describe events or actions that happened at different times, so the verbs will need to use tenses that will clearly give a sense of when the events or actions happened in relation to one another.
Take this sentence, for instance: “I’m telling you that when I met with my best friend yesterday, she claimed that she had seen your fiancée having dinner last Sunday with another man your age in that fancy bayside restaurant.” Here, “am telling” is in the present progressive tense, “met” and “claimed” are both in the simple past tense, “had seen” is in the past perfect tense, and “having dinner last Sunday” is in the past progressive tense.
In a statement consisting of several such sentences describing events or actions happening at different times, each of the sentences could have different interplays of the various tenses. The task of the writer or speaker is to make sure that such interplays of the tenses will make the sequence or progression of the events and actions unmistakably clear to the reader or listener.