We study the variation of the apparent weight of an object with height above the surface of a planet with a (buoyant) atmosphere. Interestingly, this variation depends on two competing factors--the reduced gravitational acceleration (which acts to reduce the weight with increasing height) and the reduced buoyancy force in the progressively less dense atmosphere (which acts to increase the apparent weight of an object relative to its value at the planetary surface). The relative sizes of these two competing effects are evaluated for various bodies in the Solar System. For most planets, and the one pertinent moon (Titan), the effects are quite comparable in magnitude. However, the relatively tenuous atmosphere of Mars renders the reduced buoyancy effect negligible; the apparent weight of an object thus (as generally assumed) does decrease with height. At the other extreme, the very dense atmosphere of Venus makes the reduced buoyancy effect an order of magnitude more important than the effect of reduced gravitational acceleration, so that the apparent weight of an object increases with height in the Venusian atmosphere. In addition to deriving these intrinsic results (which are admittedly somewhat esoteric in nature), studying this problem presents students at the high-school or introductory undergraduate levels with an opportunity not only to develop a deeper appreciation of the underlying physics, but also to more fully appreciate the considerable power of parametric dimensional analyses in quickly and straightforwardly obtaining (here within a factor of two) a reasonably accurate solution to a problem. The degree to which that approximate solution is sufficiently interesting (and for this particular problem it turns out that the 'sufficiently interesting' threshold is indeed met) then informs whether or not a more rigorous mathematical approach is justified.