Unlucky in love

Is it possible for a population to suffer from both inbreeding depression and outbreeding depression at the same time?

Well, unfortunately, the answer is yes, and there's an example - the Arabian Oryx.

Simultaneous inbreeding and outbreeding depression in reintroduced Arabian oryx by T. C. Marshall and J. A. Spalton in Animal Conservation (2000) 3, 241–248

(W)e found simultaneous inbreeding depression and outbreeding depression acting on juvenile survival. Outbreeding depression may be more common in vertebrates than previously supposed, and conservation strategies that seek to maximize the genetic diversity of managed populations may risk mixing lineages that are sufficiently differentiated to cause outbreeding depression among descendants.

The Arabian Oryx was driven to extinction in the wild in 1972, primarily due to overhunting. However there were a number of captive populations and reintrodtions to the wild started in 1982. The 34 Oryx reintroduced to the Arabian Oryx Sanctuary in central Oman came from 5 different populations from individuals originally captured in Saudi Arabia, Oman, and Yemen (hence the outbreeding problem). Inbreeding simply compounds this problem as the relatively 'unfit' offspring of parents adapted to different environments have to mate with one another and so problems that might remain hidden in heterozygous individuals become expressed.