We have developed a new, extremely sensitive real-space technique (intensity fluctuation microscopy) to probe the order of the nematic-smectic-A (NA) transition. Using this technique, we show that the NA transition in 4'-n-octyl-4-cyanobiphenyl (8CB) is clearly first order, contrary to calorimetric studies but in agreement with conclusions drawn from front-velocity measurements. We characterize the strength of the discontinuity at the first-order transition by the dimensionless quantity t(0)=(T(NA)-T*)/T(*). By precisely measuring t(0), we have made the first detailed tests of predictions based on the Halperin-Lubensky-Ma (HLM) theory of fluctuation-induced, first-order phase transitions. First, we explore the effect of an external magnetic field on the NA transition. Although modest fields (of order 10 T) are predicted to drive the weakly first-order transition in pure 8CB second order, we observe no such effect; we establish instead that the lower bound on this critical field is approximately 30 T. Likewise, we observe no effect in mixtures of 8CB with its longer chemical homolog 4'-n-decyl-4-cyanobiphenyl (10CB). Second, we examine the dependence of t(0) as a function of 8CB-10CB mixture concentration and find that the data in mixtures with small nematic temperature range are well-fit by the parameters derived by Anisimov et al. based on calorimetric measurements. As we increase the nematic range (by using concentrations closer to pure 8CB), the measured t(0) deviates more and more from the HLM predictions. Smectic fluctuations, which are neglected in the HLM calculation, are an obvious candidate to explain such a discrepancy, but one's naive expectation is that they would reduce t(0) below the HLM levels, whereas the observed values are too large. However, a recent renormalization-group calculation concludes that smectic fluctuations, surprisingly, should indeed increase t(0), explaining the observations presented here.