Interline agreement just means that it is possible. The airline can always refuse to do so based on policy on circumstances, e.g., separate tickets (e.g., AA/BA, etc.), separate tickets to non-alliance carriers (e.g., UA, etc.), etc. Factors like CoS and status thus come into play if part of policy or if front-line agents are willing to bend policy because of it (e.g., some UA GS have reported this).

NH staff has always checked through my bags on separate tickets at HND, but I can't speak to experiences at LAX (that's actually where I always transit).