Can you use "nevertheless" and "despite" in the same sentence?

In short, yes.

You can use the words "nevertheless" and "despite" in the same sentence.

Here is an example:

• Sometimes I sit for hours and I can't write a word. Nevertheless, despite my frustration and long hours of work, I tend to have a positive attitude toward writing in English.
• Source - Writing Clearly: An Editing Guide, 2nd Ed. by Lane and Lange, Unit 1, Exercise 4, pg 24)

In this sentence, both nevertheless and despite are both performing adverbial functions, one as a connective adverb (transition word), and one as an adverbial prepositional phrase.

Contrary outcome with connective adverbs

Nevertheless is a cause-effect linking adverb (transition word) used when there is an unexpected outcome, and it is usually located in the second clause or at the beginning of a sentence containing a contrary outcome for an opinion, belief, or action.  Its function is to show that the information in the second sentence contrasts the information in the first sentence.

In other words, in our example...

• Action (first sentence) = sometimes I sit for hours and I can't write a word
• Contrary outcome (second sentence) = I tend to have a positive attitude toward writing in English 
• (Note: This is the independent clause from the second sentence; in other words, it is the primary outcome.)

The second sentence can be broken into two parts. The word despite is part of a prepositional phrase "despite my frustration and long hours of work" that is performing an adverbial function by contributing contrasting information and describing the manner of the action in the main clause "I have a positive attitude toward writing in English."

• Manner (prepositional phrase) = despite my frustration and long hours of work
• Action (second clause) = I tend to have a positive attitude toward writing in English

The question is, is it redundant to have nevertheless and despite in the same sentence?

In this case, no. 

Yes, these two words are both adverbs signaling a somehow contrary outcome. However, they are functioning in very different ways, as they are modifying two different pieces of information.  Nevertheless is indicating a contrast between the first sentence and the second sentence and functioning as a transition word connecting two ideas at the discourse level.  Despite is indicating a contrast between two pieces of information in the second sentence and functioning as an adverbial prepositional phrase at the sentence level.

Please feel free to leave comments or questions about this explanation below.

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For additional explanations and exercises, please check out the following resources.

• Connector Review
• Connectors of Addition and Contrast (Practice Exercises)
• Despite or In Spite Of
• Quiz: Although, Though, Despite, and However