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There are many factions of Cybermen at different points in history. Most of them clearly don't have time travel, but some of them probably do—maybe pretty limited compared to the Daleks or future humans, much less the Time Lords, but enough to cause trouble.

The one clear example is in "Earthshock". The attacking Cybermen definitely either came back in time from the end of the Cyber-War, or received a message sent back in time from then, and they seem to have used their own technology to do so.

In every other example, they either have a captured timeship that they clearly don't understand how to reverse engineer ("Attack of the Cybermen"), or traveled by accident ("The Next Doctor"), or could have been taken as passengers by allies ("The Pandorica Opens").

Many fans have noticed that (Mondasian) Cyberhistory doesn't actually make sense unless you add at least one large migration into the past. But, even assuming that the writers intended it to make sense rather than just not noticing a plot hole, there's no reason this couldn't just be them falling through a time warp by accident, or another one-shot stolen device like in "Attack".

Going beyond TV to the novels, audios, etc. doesn't help. We have one story where we're told that Cyber technology is fundamentally incompatible with time travel and they'll never be able to develop it, and another where a group of future Cybermen travels back in time in a Cybership using technology that they reverse engineered from Time Lord wreckage. So, probably best to stop at "probably, some of them, but limited".