Today Superconducting Nanowire Single-Photon Detectors (SNSPDs) are commonly used in different photon-starved applications, including testing and diagnostics of VLSI circuits. Detecting very faint signals in the near-infrared wavelength range requires not only good detection efficiency, but also very low Dark Count Rate (DCR) and jitter. For example, low noise is crucial to enable ultra-low voltage optical testing of integrated circuits. The effect of detector temperature and background thermal radiation on the noise of superconducting single-photon detectors made of NbN meanders is studied in this paper. It is shown that two different regimes can be identified in the DCR vs. bias current characteristics. At high bias, the dark count rate is dominated by the intrinsic noise of the detector, while at low bias current it is dominated by the detection of stray photons that get onto the SNSPD. Changing the detector temperature changes its switching current and only affects the high bias branch of the characteristics: A reduction of the DCR can be achieved by lowering the SNSPD base temperature. On the other hand, changing the temperature of the single-photon light source (e.g. The VLSI circuit under test) only affects the low bias regime: A lower target temperature leads to a smaller DCR.