For EU flights, the answer is most likely “yes.” Under European regulation EC 261, eligibility for compensation is based on the length of delay at your final destination. If your missed connection makes you more than 3 hours late overall, you should be eligible, provided all of your connecting flights were part of the same flight reservation, under the same booking reference number.
For example the following itinerary would work (as long as you have one booking reference):
28th May at 13.00: London - New York (with United Airlines flight UA18)
29th May at 21.00: New York - Atlanta (with United Airlines flight UA69)
In addition, the compensation level is determined by your total flight distance. This includes connecting flights as long as the two flights are with the same carrier and part of the same flight reservation (under one booking reference number). This means that a delay on a short flight might entitle you to a much larger sum.
For example, even if the first flight in the itinerary below is on time, passengers are entitled to the full €600 if the London - Paris flight is more than 4 hours delayed:
28th May at 13.00: New York - London (with British Airways flight BA189)
29th May at 09.00: London - Paris (with British Airways flight BA523)
If you have any doubts or questions about your disrupted flight’s eligibility, the fastest way to check is to use the Ifdelayed eligibility check and enter your flight details here.