I do not see anything wrong with what you are going to do, however I would add a couple of rows of 2x6 blocking between the 2x6's just to guarantee that you would never have a domino effect.

You don't mention your framing situation. Stick or truss? 2x4 joists are not common in a stick framing situation. Going perpendicular to the current wall support could be an issue in a stick framed house.
If this is truss framed, I would contact someone who can confirm the wisdom of the added load.
Ron

ron6519 is right.

You also should contact your code officer or a structural professional even if the attic floor is stick built.

The reason, again, is your existing attic floor joists are likely not able to bear the added loads from the 2x6s, the plywood and items you wish to store on them.

The new attic floor system you describe will support about 18 psf Live Load and 7 psf Dead Load which is just slightly below the usual 20/10 design load for attic storage accessed by ladder. Be sure the walls you are connecting to are bearing walls and the connectors are adequate.

But the greater issue is deflection which would be about L/270 with Hem-Fir lumber and might allow the ceiling to crack if you started crawling around up there. Building codes like to see L/360.

If you used 1 3/4 x 5 1/2 PSL's or LVL's with an E of 2.0, the deflection would be close to the code required L/360 at a loading of 20L, 10D and the original ceiling joists might help reduce the deflection as well but you didn't mention how far they span or the kind of ceiling.

Doubling the 2x6's (or spacing them at 8" o.c.) would be even better.

I don't think the OP is reading this anymore... but I don't want others to get possibly false ideas...

Not a structural engineer, but I do have some ideas of basic principles... I'm confused by macv's design load remarks...

How can we take a ceiling - which is NOT designed to carry a load, and is only designed to act as supports for drywall, and help distribute roof loads from the trusses...

And then take these 2x4's and convert into a FLOOR - just by adding some 2x6's perpendicular???

Your not actually working on a FLOOR. A ceiling is a floor - when it is DESIGNED to be a floor. It has load bearing walls beneath it; or pillars - at specified points - to transfer the expected loads to the foundation.

An attic ceiling which is not designed to carry a load - cannot be created into a proper load bearing floor - just by building on top of it...

Otherwise - I could have Cardboard for a ceiling - and then just carefully add some 2x8's on top of the cardboard, etc, and I'm good to go... NUTS, not!

He stated the 2x6's would have a span of 14'. He never stated what the span of the 2x4's was...

A framing book that I own, doesn't even list 2x4s as floor joist span ratings - the chart starts with 2x8's...

So I just think that a couple of the replies above are extremely misleading. I totally agree with everyone who said contact a structural pro or building code person... Course all of those individuals are going to state: Sorry - can't be done, unless you want to major remodel..

A foundation of 2x4's - does not work as a FLOOR. And as manhattan mentioned - the 2x6's and plywood, are just adding more load on top of the foundation of 2x4's... You would have to have a tremdous amount of load bearing walls, or posts beneath in specified locations - to enable the attic 2x4's to be able to properly transfer a load...

The OP is planning to add attic floor joists spanning from one wall to another independent of the ceiling joists therefore the strength of the ceiling joists would be ignored.

"14' pieces would run wall to wall in the section I'm flooring"

"Be sure the walls you are connecting to are bearing walls and the connectors are adequate."

I have the same situation but the reverse. I have currently 2x6 and i was planning to raise the floor by putting 2x4 across. Should i do that or put directly on top of the 2x6 going the same way?