It's hard to put an exact date on it, but I think there's a lot of argument for the period 1867-1901 being time period in which this happened. Although Austria-Hungary never became fully impotent and better command at the beginning of World War One could have prevented the embarrassing defeats in Galicia and Serbia, this period did mark her decline from the top rung of the Great Powers to being one of the weaker ones. Austria-Hungary's army became smaller, recruiting only 1 man in every 132 (compared to 1 in 98 in Russia and 1 in 64 in France), meaning that despite having the 3rd largest population of the European Great Powers in 1914, her army was smaller than all except Italy's. Economic backwardness explains some of this, as despite the great progress made at the turn of the century Austria-Hungary was still poorer and less industrialised both in total and per-capita when compared to Britain, Germany and France, though not necessarily to Russia and this is where the impact of Austria-Hungary's "peculiar institutions" show themselves.
The army was poorly funded in part due to the intransigence of the Hungarians in particular, though Franz Josef was feckless enough to allow his empire to atrophy. In addition those in command of the Empire's army in 1914 were very much inexperienced among other things, allowing their forces to be repeatedly defeated by the Serbs, who were materially inferior to the Austro-Hungarians in nearly any way you can imagine.