Deep learning-based weather forecasting (DLWF) models have recently demonstrated significant performance gains over gold-standard physics-based simulation tools. However, these models are potentially vulnerable to adversarial attacks, which raises concerns about their trustworthiness. In this paper, we investigate the feasibility and challenges of applying existing adversarial attack methods to DLWF models and propose a novel framework called FABLE (Forecast Alteration By Localized targeted advErsarial attack) to address them. FABLE performs a 3D discrete wavelet decomposition to disentangle the spatial and temporal components of the data. By regulating the magnitude of adversarial perturbations across different components, FABLE produces adversarial inputs that remain closely aligned with the original inputs while steering the DLWF models toward generating the targeted forecast outcomes. Experimental results on real-world weather datasets demonstrate the effectiveness of FABLE over baseline methods across various metrics.