Yes, it's possible, but it depends on the benefit and your willingness to wait in line.
For example, no airline is going to award you bonus miles based on your elite status in a different airline's program. But for other services, you can game the system a bit if you are willing to wait in line to have your reservation switched from one number to another. For instance, you might reserve the flight with your status account, and use it to benefit from priority check-in, priority luggage tagging, complimentary upgrades, or lounge access as may apply. Then, you can have it changed at the gate to your accrual account prior to departure. If you want to use the lounge at your connection, you'd probably need to have it switched back, then switched up again.
Many airline reservation systems do allow multiple frequent flyer numbers to be entered for a passenger. Your main FF number is generally entered as the FQTV (FreQuent TraVeler), and the account to be used for mileage accrual. They may also record a separate FQTS (FreQuent Traveler Status) number for the account on which your status and thus benefits are based. (Other values may include FQTR and FQTU; you can search on those values, for instance, in this 2009 Amadeus manual; there are also many FlyerTalk threads on FQT*.)
In practice, things are not quite so simple as ringing the airline and having the second number added. First, even in what I would consider the golden age of frequent flyer programs of a decade ago, I found it hard to find an agent who knew what I was talking about, was willing to add the entry, and was able to enter the numbers correctly. Second, on itineraries with multiple airlines and/or codeshares, it should be no surprise if one or the other number gets lost. Third, whether it is the FQTV or FQTS gets printed on your boarding pass and in the manifest may vary, which may affect what benefits you are able to enjoy in practice— a lounge agent might reject you based on eyeballing your non-status FQTV printed on the BP, and you might be passed over in meal selections based on your FQTV on the manifest (for airlines which order meal choice by status).
It is not generally in an airline's interest to make it easy— you are putting them on the hook for supplying you with benefits when you are benefiting, and possibly switching to, a competitor. No airline I'm aware of publicizes the service, and there are various anecdotal reports of restrictions on specific airlines— that Singapore only allow Krisflyer FQTS, or that it's no longer possible on United. It was not reported impossible of BA as of October 2014, for what it's worth.