I'm guessing this must have been somewhere around the beginning of the colonial era? (Mid 16th century?) Earlier? Are there a lot of failures of European armies in that era that don't generally make it to common knowledge?
I don't really understand this question. I assume you are referring to when Europeans started colonising the world but there are many 'non-european' armies that they didn't properly defeat or maybe wouldn't even be able to compete with.
The Ottoman empire's army in the 1500's was one of the best on the planet and the armies of Japan were also pretty good when they were unified. China could call on hundreds of thousands of troops.
Colonialism was successful because the countries they targeted initially where not countries at all. India wasn't unified at all, Most Africa 'countries' below the Sahara didn't have a concept of nationhood yet. The ones that did weren't as easily conquered if at all in some cases.
As for failures Europeans were pretty hopeless against Muslim empires in the medieval, early renaissance. Whether it was Spain or the disaster of basically every crusade after the third one. Probably the most famous modern Colonial European loss is when Japan beat Russia in 1905.