CX is the final authority in determining whether they can check your bag through with DL on a separate ticket. The reason is simple, the experience we share with you, even if exactly comparable, may be outdated. Airlines don't always have baggage agreement with each other and lately, many airlines like DL and BA have adopted baggage policy which they will not tag through bags from two separate tickets unless they are from their respective alliance. These agreements do fluctuate and change over time. Somebody who had a similar experience in early 2015 and told you yes but in fact it's a no when you show up at HKG.

If you are flying DL back to JFK to connect to your CX flight, I know for a fact DL will not tag your bag through to CX. I can say that with confidence because my information is one week old and DL explicitly told me that they will no longer tag through bags from anyone other than Skyteam airlines unless it's on the same ticket.