Paragraphs are structured as self-contained thematic units, where all of the sentences in a paragraph should deal with the same topic. Any supporting sentence that stray off-topic do not belong in the paragraph and needs to be "supported" by a different kind of topic statement. The only sentences in a paragraph that are not supporting sentences are the first (the topic sentence) and the last sentence (the concluding statement, which summarizes what you have just said).
For instance, "I had a great time in New York" serves as an adequate "topic sentence" to open a paragraph. Some appropriate supporting sentences for this opening may include: "I visited the Empire State Building and had a picnic with my friends in Central Park. The weather was glorious for the whole week. On the last day, we saw the Statue of Liberty and went to see a Broadway show. The flight back was so peaceful and hassle-free that I slept most of the way."
The topic sentence in the preceding example presents a clear opinion (that the holiday was "great") to the reader. The remainder of the paragraphs breaks down why the holiday was great -- because the writer saw his friends, visited several tourist attractions and had an easy journey home. The sentences "support" the topic sentence by providing ample evidence as to why the holiday was great.
If a paragraph opened with "I had a great time in New York," but the following sentences read: "I've been thinking about leaving my girlfriend..." or "I'm not sure I can stand my job for much longer; my boss is a nightmare...," they would not "support" the paragraph opening in any way. They would just drift aimlessly to a new, unrelated subject area. As readers, we have no idea why the holiday was great, as there has been no expansion on the subject. For this reason, they would not belong in the paragraph, as they offer nothing in support of the initial statement. These sentences would be much more at home in a paragraph that began "Life has been rough recently; I think I need to make some big changes."