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I just watched S7E25 "All Good Things ..." of Star Trek: The Next Generation which I believe is the final episode of the series. In this episode, Captain Beverly Picard tells the crew to pilot her medical ship into the Neutral Zone at Warp 13. In a separate episode of ST: Voyager (actual episode unknown to me), the characters Tom Paris and Harry Kim talk about not being able to break the Warp 10 barrier for some reason. This is toward the end of the series. Is there any explanation in the Star Trek Universe which explains this seeming difference? This especially confuses me because ST:V (1995-2001) started the year after ST:TNG ended (1987-1994). I would think it would be produced with these story lines in mind. Is there any explanation for this?

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The episode in question , "All Good Things" (TNG: S7E25) is actually based in an alternative universe. Warp factors above 10 do not exist according to 24th Century Warp Theory as "Warp Factor 10" is interpreted as "Infinite Velocity".

It is important to note the differences between the earlier Star Trek material and the later material. Many episodes in The Original Series quoted warp factors that were later 're-scaled' as the newer series (The Next Generation, Voyager, Enterprise) came into existence. This is covered in more detail on the Memory Alpha for Warp Factors.

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That link is a treasure trove from the Star Trek Universe ... it also spells out all the inconsistencies brought to light during all the series about how fast warp speed is, lol. – Pᴀᴜʟsᴛᴇʀ2 Oct 8 '13 at 15:47

This has been answered in some detail on SFF. The answer references the Memory-Alpha wiki for All Good Things … where the following is noted:

It is clear that there is no more warp 10 limit in the future. This limit was set in "Force of Nature". In addition, ships in the future timeline are able to go above Warp 13. This would appear to contradict "Threshold", where it is stated that Warp 10 is the theoretical limit. It is, however, possible that the warp scale was recalibrated in this future.

In other words, the limit is exactly the same. It's just the labels on the scale that (might) have been changed.

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