LOL I'm certainly not any more (and likely less) qualified to address this than are some others on this board, but I'll give it a shot.
Inversions can be the result of many different processes. For example, during the evening and early morning hours, it's quite common to have strong radiational cooling at the surface lead to a low-level inversion (called a radiational cooling inversion, for obvious reasons). Another source for inversions is the result of subsidence aloft. For example, DNVA (different negative vorticity advection) aloft can, per the QG equation, lead to subsidence aloft. Such inversions are often marked by not only increasing T with height, but also by a sharp dry layer in the inversion layer (e.g. you'll often see a rapid drop in Td/RH in the inversion layer). It's worth remembering that inversions are defined as layers in which the temperature increases in height. Such layers are always stable. However, it's also worth noting where lapse rates are absolutely stable (approximately less than 6C/km), which can still provide significant CINH for a potential parcel attempting to rise through the layer.
Getting back to inversions (though we probably should note stable layers, not exlusively inversions)... Most of the time, the radiational inversion source is not going to be present for daytime convection. The few times that radiational inversions do linger through the following day, in my experience, tends to occur when the previous day had a very deep boundary layer, and the current day does not mix through that depth (e.g. a day after a dryline passage may allow boundary layer / mixing depths to 700mb or above, while the next day may not mix to that depth owing to increased low-level cloud-cover, etc). Subsidence inversions, however, can always be an issue. Oftentimes, we see the development of subsidence inversions behind departing waves (where DNVA and perhaps ageostrophic curvature convergence aloft results in subsidence aloft), as was experienced on one of the busted MDT risk days in IA a few months ago. This is part of the reason why tracking shortwave troughs on watervapor can be very important when judging the potential for initiation.
Per the QG approximation, subsidence also results from cold-air advection. Yes, it's rather interesting that a response to cold-air advection is subsidence, which is a warming process. Strong CAA in the mid-levels may not necessarily lead to the development of an inversion aloft, but you could see synoptically-significant downward motion. Stronger vertical motion will occur when CAA takes place in a layer with high lapse rates than in a layer that is more statically-stable. Remember, air "travels" on isentropical surface... Motion will be in the vertical when the isentropic surfaces are vertical (e.g. a dry-adiabatic lapse rate signifies that the potential temperature is constant in the vertical); the higher the static stability, the less vertical motion that will result.
Warm-air advection just off the surface, though resulting in upward motion, can result in stable layers as well. Such absolutely-stable layers are common north of warm fronts (in North America, assuming warm air to the south), with the stability often enhanced by the presence of cloudcover and/or precipitation associated with isentropic upglide. Such inversions are called 'frontal inversions', and can be found behind cold fronts as well (but we don't chase often behind cold front).
Stable layers above a given location may also result from the quasi-horizontal advection of warmer air from upstream convection. For example, suppose there an area of vigorous convection to the west of a given location, with easterly mid-level flow. Updrafts often are (but not always!!) characterized by a positive temperature perturbation and moist-adiabatic lapse rates. These warmer temperatures aloft may be advected with the mean flow, and affect the temperature profiles downstream, even after the convection has dissipated. Fortunately, it's most common to see the nocturnal convection dissipate during the mid-morning hours, providing enough time for that convectively-affected air to be advected downstream (and hopefully away from the target area). With time, diffusion/detrainment/turbulence will act to reduce the intensity of this area as well...
I'm not sure about the earlier busts having multiple inversions. I've seen multiple strongly-stable layers on many chase bust days before as well, many of which featured a mid-level subsidence inversion (or two subsidence inversions) atop a strongly-stable layer. Although synoptic-scale vertical motion associated with DVA and thermal advection is often small (and several orders of magnitude less than that associated with moist, deep convection), my experience has shown that it can have a significant influence on initiation. The 2002 chase season pops into mind whenever I think about large-scale subsidence leading to busts, as we had several decent events "ruined" by shortwave troughs that progessed through the target area in the morning, resulting in DNVA and associated subsidence in the area by afternoon. All of this is purely anecdotal, but I'd much rather be downstream from a trough axis than upstream from it!
Anecdotally, I think the multiple stable layers (again, not necessarily inversions, but layers with lapse rates less than the moist adiabatic lapse rate, which will provide CINH for parcels that have a lower potential temperature than that of the layer) tend to be less of an inhibitor on days with high-quality low-level moisture. If you want to breach through stable layers with as little CINH as possible, you want to shift the parcel trace as far right as possible, which means having parcels use latent heat as low to the ground as possible. Higher low-level moisture will yield higher theta-e (all other things constant), which will lead to a rightward shift in the parcel trace. Generally, however, the higher the static stability (like an inversion instead of an isothermal layer) in a particular layer, the less of an affect a small increase in parcel theta-e will have in reducing the CINH. If the CINH profile is long-n-slender (often resulting from a layer with a lapse rate just weaker than moist adiabatic), a minor increase in parcel q may result in complete removal of all CINH. Note also that, as low-level moisture increases, the importance of the vertical temperature correction to the projected parcel trace increases as well (assuming you want to visualize the parcel trace to estimate the CINH/CAPE). In addition to high-quality moisture, I also look for strong, sustained deep-layer convergence. Deep-layer convergence can lead to upward motion through a capping layer, which may cool that layer (as can upward ascent ahead of a trough axis / vort max).
EDIT: Regarding yesterday's soundings... The 00z MPX sounding has a nearly-saturated weakly-stable layer below 700mb (with near moist-adiabatic lapse rates), but nothing in the way of an inversion. There was nearly 3000 j/kg CAPE for a parcel originating at 834mb on the sounding, which makes it no surprise that significant elevated convection developed. The GRB does show multiple inversions, one in the 750-800mb layer and one in the 875-900mb layer. There's another, though suspcisiouly shallow, strongly-stable layer near 690mb that appears to be an isothermal layer rather than an inversion.
EDIT2: This post is WAY too long, so I apologize. Once I get typing, I can't stop! Most of this post addresses material that you didn't necessarily inquire about, so consider most of this a digression.
