If they've been in the ground for more than three years, I'd dig up a spade full and check them. The "sucker" bulbs you describe are just developing bulbs and if there are very many in the spadeful, then I'd recommend digging and separating. Keep only the viable ones... i.e., ones that are at least one inch in diameter. Discard the smaller ones. Brush and store as related above and plant in the fall... We plant ours here in the U.S. in October. They should be planted at least 5 to 7 inches deep and at least 8 inches apart. For something unique, try planting some tulip bulbs as described and then, over the top of these plant some crocus bulbs. The crocus will come up and bloom first then followed by the tulips... Good luck!