Can you have a <span> within a <span>?

Yes. You can have a span within a span. Your problem stems from something else.

HTML4 specification states that:

Inline elements may contain only data and other inline elements

Span is an inline element, therefore having span inside span is valid. There's a related question: Can <span> tags have any type of tags inside them? which makes it completely clear.

HTML5 specification (including the most current draft of HTML 5.3 dated November 16, 2017) changes terminology, but it's still perfectly valid to place span inside another span.