Precision pulsar timing can be used for a variety of astrophysical tests, from the detection of gravitational waves to probing the properties of the interstellar medium. Here we analyze various noise contributions to pulsar timing residuals from continuous multi-hour observations of seven millisecond pulsars (MSPs). We present scintillation bandwidth measurements for all MSPs in the sample, some for the first time. We also present scintillation timescale measurements and lower limits for all MSPs for the first time. In addition, we present upper limits on the contribution of pulse phase jitter to the timing residual error for all MSPs. These long observations also allow us to constrain variations in dispersion measures (DMs) on hour-long timescales for several millisecond pulsars. We find that there are no apparent DM variations in any of the MSPs studied on these timescales, as expected. In light of new radio telescopes, such as the Canadian Hydrogen Intensity Mapping Experiment, which will be able to time many pulsars for a short time each day, we search for differences in timing precisions from continuous pulse times of arrival (TOAs) and from equivalent length time-discontinuous TOAs. We find no differences in the precision for any of the MSPs in our sample, as expected. We conclude that the TOA variations are consistent with the expected breakdown into template-fitting, jitter, and scintillation errors.