This is not an error that is very often discussed in relation to altimeters, though it can exist, particular with older aircraft on mechanical instruments (so called "steam gauges").

The altimeter is usually a very precise and fast reacting instrument, particularly when compared to its cousin, the VSI (Vertical Speed Indicator), which takes a few seconds to start indicating a rate climb/descent due to its "metered leak" principle of design (not the same for an IVSI - Instantaneous VSI).

Altimeters can have an associated lag, though, as it takes just a little bit of time for the pressure change to run through the system and reach the gauge (this is barely noticeable but you may be able to see it yourself in practice). Primarily, though, the most noticeable lag on an altimeter occurs when altimeters have had the same pressure for a long period of time, such as in the cruise or on the ground, and the diaphragm therefore has a certain amount of "mechanical inertia", and they take extra force to get them moving from that position, which takes a little time to accumulate when starting a climb or descent. This is why more advanced altimeters are fitted with tiny vibration motors to avoid any such issues as the diaphragm/bellows are always moving ever so slightly.

There are other little errors possible such as altimeter leading/lagging in rapid climbs or descents, but these are come across less often.