I'm still not sure if Lawness is asking if a necessary condition and a sufficient condition can happen simultaneously (meaning that if these things were really happening, could they happen at the same time or does one have to come first), or if (s)he's asking if a necessary condition can also be a sufficient condition and vice versa--but here's something that might help regardless. Logically speaking, 'necessary' and 'sufficient' are pretty much technical terms that refer to the logical relationship between two sentences/ideas, rather than to the actual relationship of whatever is being discussed. In other words, here's a sentence:
If I have gas in my tank, then my car will run.
Even though in real life, it would be more accurate to say that gas is necessary for my car to run, in that sentence 'gas in my tank' is the sufficient part and 'my car will run' is the necessary part. This is because 'necessary' just means "this is necessarily true if the other part is true"--which is what the conditional means. And 'sufficient' just means "the truth of this statement is enough to guarantee the truth of this other statement". The two statements that make up a conditional don't have to have any sort of realistic relationship at all--I can slap any two statements or ideas together and form a conditional, and one part will still be the sufficient part and the other the necessary part--so it might be easier not to think about how the statements are related to eachother in a real-life sense--because all any conditional really means is "if this is true, then that is true" -- why or how they are related to each other doesn't matter at all.
I was really asking if they can occur in real life terms simultaneously because I was trying to make sense of the two with a real world example. It helps to think of the two instances/events in logical terms and without subscribing meaning to them. For diagramming purposes, the sufficient condition ALWAYS comes first, right? At least that is what LRB says.