I am not a fashion historian (except for fashion in metal), so I can't compare the rapidity of changes in armour fashion to the rapidity of changes in civilian fashion.

With that said: Armour fashions absolutely changed decade-to decade during the 'age of plate armour' - the 14th through the 16th centuries. In the 14th century armour developed rather rapidly as plate armour transition from a supplement to mail armour (in the first part of the 14th century) to an independent body defense. This meant that new defense for the limbs were added, existing defenses were tinkered with and made more extensive, and the form of plate armour changed through the century. In this environment of technological change, 'updating' your armour would be the difference between having the latest in life-saving technology and having something that was merely pretty good. To take a late example, in the early 15th century more and more helmets had a metal defense for the throat (this comes in a variety of forms, but the most common is the Great Bascinet). This offered better protection around the throat than the mail aventail on late 14th century bascinets. Now, if you were a French man at arms at Agincourt, a plate defense around your throat might be the difference between life and death when you were shot at by dozens of arrows! Arrows that would probably bounce off your plate armour, but might (might!) penetrate mail. Other developments in armour make it more maneuverable and better adapted to fighting in dfiferent circumstances. For instance, in the mid 15th century the afformentioned great bascinet was replaced throughout northwestern Europe by the sallet (here is a somewhat unusual late example) which offers better mobility of the head and better visibility. In the previous century the heavy great helm had been replaced for most purposes by the bascinet. By 1420 plate armour encompassed the entire body (the last piece to develop was the backplate, which until the early 15th century was made in multiple pieces even when the breastplate was one piece), so further improvements weren't matters of protecting new parts of the body but tinkering with armour's form. Now, this is not best thought of as a linear improvement. Warfare itself changed over the Middle Ages, and different nations tended to fight differently. The English men at arms went from fighting on horseback to fighting primarily dismounted, and their armour was adapted do this. A lot of updates to the form of armour can be seen as refining the form to meet the function - either improving an existing function, or adapting when the function has changed. Finally, in the 15th century the metallurgy of armour improved as carbon content increased and heat treatments became more common (at least in Italian and South German armours, there is not enough evidence to say what heat treatment was used elswehere). Put all this together and newer armour could be both more protective and more comfortable. Did this mean that older armour was truly obsolete? No, it was not - it would still protect, even if it protected a little less and was a little uncomfortable. Wearing armour from 30 years prior in the Wars of the Roses was not like a soldier from the American Civil War trying to fight in the Boer War, to use a modern example - the devleopment of the technology just didn't work like that dramatically.

But that's not everything. Fashion absolutely plays a role in armour history. Armour reflects the clothes that people wore in a variety of ways. The neckline of breastplates follows the neckline of men's gowns hence the squared-off neckline of early 16th century German armours. The shape of sabatons, the waistline of armour, the silouette of the armour, all of these reflect ideas about what the male body should look like. Armour that didn't match this idea would look as out of place as wearing your dad's blazer from the 70's with it's four-inch lapels.

Add up the changes due to function and those do to fashion and armour changed relatively quickly. Not anything like today's fasht-fashion world, but still noticeably. Even in times when armour seems to change little, like the period 1420-1450 in England, Tobias Capwell was able to break armour fashion down by decade, based on the armours depicted in the effigies of men at arms in churches (these are monuments created to prompt people to pray for the soul of the dead). At other times, such as the period from 1480 to 1510 in Germany, armour styles change dramatically. Armours go from the distinctive 'gothic' style to a more 'rounded' style that superficially resembles the armours of Italy in many respects. The cause for this sudden change was the 16th century Northern European love of all things Italian, which can be observed in everything from Durer's journey to Italy to the adoption of dueling culture to the names and settings of characters in Shakespeare's comedies. In this case the change in the form of armour reflects a larger cultural trend.

It is worth keeping in mind that armour was expensive, yes, but full harnesses (suits) of armour was worn by the richer echelons of society - the elite if not the full-blown aristocracy. And it was worth spending money (money that they had!) on keeping relatively 'up to date'. Among those with means, buying an armour that was customized and fitted to the wearer would be desired - no reason to fight in your father's harness when you could buy one that fit you precisely. However, keep in mind this only applies to the upper echelons of the military elite - mostly those with titles - most men at arms would not wear truly custom armour. Speaking of those lower in the hierarchy*, they would often receive armours as gifts from their social superiors - knights serving a lord would receive harnesses, as might men at arms serving a knight. Late medieval and early modern plate armour was valuable, but it was not a priceless heirloom passed down from father to son. Thus men at arms had both motivation and opportunity to wear newer harness.

All of this means that armour was absolutely subject to fashion and to technical changes, and people kept up with fashions, at least to an extent. But this is not to say that old armour was not worn. On the contrary, we have evidence from effigies in the first half of the 14th century that men at arms wore differning degrees of plate armour - some men at arms continue to not wear any visible plate armour well into the 14th century, while other have adopted noticeable limb protections. This may reflect older equipment being retained, perhaps by older men at arms. Later on there is one curiously old fashioned armour in an effigy from the early 15th century, that may reflect a more 'conservative' set of armour worn by an older man at arms, or may reflect the armour he wore at Agincourt. Moreover, we know that there was a second-hand trade in armour - some merchants would even arrive after battles and purchase the armour of the defeated! Armour was an important part of the booty that medieval armies collected after (admittedly rare) pitched battles. This armour would be sold and either recycled (reworked into metal plates, perhaps, or into the small plates for brigandine armours) or re-used. Armour is also inventoried in will, so it was inheritted. However it wasn't necessarily passed on to sons. Remember those gifts of armour? Magnates might gift their old armour (either on their death or while they were alive) to their retainers. This second-hand armour would likely be re-worked for fit in some way (though it would never be as comfortable as a custom harness), and might be used for some time long after manufacture.

Thus is a sense A Knight's Tale is plausible in showing older armour being worn and in it being something that might be remarked upon.

This answer is indebted above all other sources to Tobias Capwell's Armour of the English Knight 1400-1450.

*(and this is a relative term for people that are still in the upper 10% of the population)