Welcome.

Since you are new, you may want to do a Search on some of our comment for additional education.

Ok you need to clarify... "It will kick in but not turn over" Does that mean the engine is cranking but it will not fire off OR does it mean the Starter trys to turn the Engine but the engine will not even crank?

If it's that the engine is cranking but wont fire up then:
this intermittent no-start usually is a few different things.

**The Crank positioning Sensor is going bad. (http://www.fcpgroton.com/product-exec/product_id/3396/nm/1989_1993_Volvo_240_Engine_Speed_Sensor_Impulse_Sensor/category_id/116) Located in back of engine, mounted into the Bellhousing of the transmission.

**The Fuel Pump relay might be going bad. It can be opened and resoldered. It is a white relay located on the passenger side of the car behind the glovebox. You can get to it by removing the Knee bolster (padded thing).

**Fuel pump. they do get tired I replaced mine at about 170K miles. If the car doesn't start AND you did not hear the Pumps run when you turn the key to the 'ON' position, tap the Fuel pump with a hammer and see if the car starts. FuelPump is located under the car just behind the drivers seat.

Is this No Start probelem getting worse over time or is it about the same frequency as when it first started?

I would start there.
Do you do your own mechanical work on the car or are you looking for ideas before getting beat up by a Shop?

--
'75 Jeep CJ5 345Hp ChevyPwrd, two motorcycles, '85 Pickup: The '89 Volvo is the newest vehicle I own. it wasn't Volvos safety , it was Longevity that sold me http://home.no.net/ebrox/Tony's%20cars.htm

Have I read this right - the starter motor engages but it won't turn over?

That can only be a starter or ring gear or big cable problem.

Look for frayed wire/insulation on the crank position sensor mounted on the top of the bell housing behing the head. If insulation is failing replace the sensor.
Fuel pump relay will also cause the engine to stall or no start.
Dan

Welcome to the brick board : )

Have you checked your fuses, namely fuse number six?
Also, can you hear your fuel pumps turn on when you turn the key?

You might also want to check the diagnostic system for codes, there are instructions in the 700/900 faq.

Good luck,

Justin