They are both written as 何時 (what time) in the dictionary but they seem to be different words.
My understanding is that "なんじ" asks for "exact hour" while "いつ" is like a general "when". Is that right?
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In short, yes.
何時 (なんじ) asks for the time. It is analogous to 何日, 何月 and 何年.
「何時 (なんじ) に行きますか。」
いつ is general, and includes 何時, 何日, etc. Usually it is written in hiragana.
「明日にしましょう。」 or 「10時15分にしましょう。」