求解释: "我有问题" 和 "我有个问题" 中英 | Chinese & English
It's actually very common to say “我有问题”, and indeed it would cause ambiguity without contexts.

I think it's because 问题 could be both "question" and "problem". The ambiguity can only be cleared within the context.

Although ambiguous usage should be avoided when it's likely to create confusion (actually, eliminating ambiguity in sentences is a vey common practice on language classes in China, but it's not a very strict grammar requirement), this kind of situation is very common in Chinese because the language is an Analytic Language which depends heavily on contexts where English depends more on the inflection of words and grammar.

This reminds me another similar situation when I was learning English. I thought "I'm hot" in English technically could mean "I feel hot", but it seems that to avoid ambiguity, most people probably would say "it's hot" or "I feel hot".