Sounds like breakdown in insulation in the ignition side of things - could be the low tension or high tension side.
Check all the wires and clean them up. Any cracks or splits replace them.
Check the plugs, check the white insulator at the tip isn't cracked or broken.
If the engine doesn't mis-fire before "bogging" then pay close attention to the lead from the coil to the distributor.
Also check all the earth straps for the engine.
What engine and ignition system is in the car ?
There are a few voltages and resistance you can check if it's not a complex ECU. If it's all in the ECU brains, take it to a garage and get them to plug it in to the diagnostic computer. 30 seconds will find the problem.
OTHER things it could be as stated, fuel filter, most likely if it hadn't been for the wet making it worse.
Is the tank a closed system ? If it's open, make sure the breathers aren't' blocked. An easy test for that is to run it with the cap OFF. If it runs fine then the tank is becoming a vacuum and preventing fuel flow.
How quickly is it bogging down ? Could be crankcase pressure. To be that bad it would be obvious that it was pressurising by feeling the oil filler opening with your hand.
The other (worse) is it coull be rings/bores as the oil thins it is less able to maintain the seal. But again. not somethign you'd expect worse in the rain
"A woman without curves is like a road without bends, you might get to your destination quicker but the ride is boring as hell'