Peony buds that turn brown and fail to open are symptomatic of a disease called botrytis. This fungal disease is worse in cool, wet conditions, so do what you can to increase air circulation around the plants to promote quick drying. Space plants far enough apart that air can flow around them. Water only at the base of the plants to avoid wetting the foliage and flower buds. Prune or remove nearby trees and shrubs to increase the amount of sunlight and wind that will reach the plants. Fungicide sprays may also be used to prevent development of the problem. Apply the first spray when the peony shoots begin to emerge, and repeat at 10-day intervals during humid conditions or until the plants stop blooming.