The Short Version
Pickled cucumbers absolutely can and do spoil. The National Center for Home Food Preservation (UGA) explicitly states pickled products are "subject to spoilage from microorganisms, particularly yeasts and molds." Opened jars last roughly 3 months refrigerated, and even unopened jars have a finite shelf life of 1-2 years. While vinegar slows spoilage significantly compared to fresh cucumbers, it does not prevent it indefinitely. Signs of spoilage include mold, off odors, mushy texture, fizzing brine, and bulging lids.