The phrase "but also often" is correct and usable in written English.
You can use it to introduce a contrasting idea or point. For example, "The weather is usually sunny here, but also often rainy."
Work came quickly but also often predictably.
His new life is happier and more hopeful but also often strange.
"This character makes a lot of dumb decisions but also often has the perfect one-liner.
It's complicated: What keeps her isn't just fear, but also often an emotional connection.
He could be hilarious company, but also often outrageous and childish.
The policy is not only laid down by the president but also often has enthusiastic support at state level.
The costs are great, not only for the targets of anti-gay efforts but also often for the perpetrators.
His brilliant wit was often admirably used against fads, affectations, and muddleheadedness but also often cruelly and spitefully against personalities.
The pictures are meant to help people find their way around, but also often show passers-by in embarrassing situations.
The deeper the water, the further the exceptions stretch, not just from federal guidelines but also often from company policy.
This is a medium in which the past is not only a foreign country but also, often, an unreachable one.
