We focus on systems that naturally incorporate a degrading quality, such as electronic devices with degrading electric charge or broadcasting networks with decreasing power or quality of a transmitted signal. For such systems, we introduce an extension of linear temporal logic (Linear Temporal Logic with Degradation Constraints, or DLTL for short) that provides a user-friendly formalism for specifying properties involving quantitative requirements on the level of degradation. We investigate the possibility of translating DLTL verification problem for systems with degradation into previously solved MITL verification problem for timed automata, and we show that through the translation, DLTL model checking problem can be solved with limited, yet arbitrary, precision. For a specific subclass of DLTL formulas, we present a full precision verification technique based on translation of DLTL formulas into a specification formalism called Buchi Automata with Degradation Constraints (BADCs) developed earlier.