Define "human-like sounds"...
Various African languages use click sounds. Most European languages lack the two "TH" sounds from English. English itself lacks the French "J" sound, and English English lacks the Scottish "CH" sound. French and Finnish both differentiate rolled and unrolled "R" sounds. Japanese has one consonant halfway between "R" and "L" which causes many Japanese to be unable to distinguish the two as separate consonants when speaking or listening to European languages. And then we get onto tone languages like Mandarin.
More unusually, there are even whistle languages.
And that's just for formal language. The human voice is capable of some remarkable sounds, as demonstrated by beatboxers purely for musical and rhythmic effect.
Children are born capable of adapting to all these sounds, and all adults continue to be physically able to produce these sounds (albeit sometimes requiring practise, and demonstration from other people of how to produce them). However during early development, babies lose the ability to hear vocal sounds other than those around them, so that by the time they reach the "babbling" stage, the sounds they produce are only those sounds which they have already heard from people around them.
Given the diversity of sounds possible from the human voice, a similarly capable alien language may have a similar diversity of sounds. The problem you have then is explaining why the aliens only have a single language! (The old Star Trek problem of "one village represents the whole planet".)