The phenomenon you describe, where the same result comes about in more than one time/location via evolution, is called "convergent evolution." It is when the same character, or trait, evolve independently of one another. One example is the wing, it has evolved (at least) several times independently, in bats, birds, and insects. Parallels can also be drawn with other traits like dispersion methods of seeds, or the evolution of fins in fish, dolphins, and whales. Here is a classic paper by Ernest Williams from 1972 which describes one of the earliest examples given of ecomorphs, the anolis lizard, also talked about on this site.
This is from a good article on the new scientist website that would be worth reading:
"Evolutionary convergence occurs at every level, from proteins to
societies. An unusual antibody once thought to be unique to camels has
a close equivalent in sharks, for instance, while naked mole rats form
social colonies like those of ants and bees.
"What this means is that if we could wind the clock back and let life
evolve all over again, life might take very different paths but still
produce organisms that, in some ways, resemble the organisms alive
"There would almost certainly be streamlined swimmers in the oceans and
winged creatures in the skies. In fact, some argue that the evolution
of intelligence is also virtually inevitable, though intelligent
organisms could be very different from us."
The answer to your question is yes, with enough time and the same selection effects we would not be surprised if there was a similar pair of independently evolved characters. However, selection acts on random mutations which is not a directed process, therefore for the same characters to evolve you would need mutations to occur that had similar effects. This is why given enough time we can see the same responses. Despite the non-random effect of selection, we could consider evolution to be a stochastic process, it only arrives at the same converged point by chance. This is because selection (the main driver of evolution) acts on randomly generated genetic variation thus the direction of the evolutionary trajectory is random.
"..often used to represent the evolution of some random value, or
system, over time. This is the probabilistic counterpart to a
deterministic process (or deterministic system). Instead of describing
a process which can only evolve in one way (as in the case, for
example, of solutions of an ordinary differential equation), in a
stochastic or random process there is some indeterminacy: even if the
initial condition (or starting point) is known, there are several
(often infinitely many) directions in which the process may evolve." -
wikipedia entry on Stochastic Processes
Following the quote above it may seem that evolution is a directed process, i.e. selection is aiming for a specific target. It is not, selection can only weed out the least fit genotypes and increase the frequency of the better ones, selection has no goal or pre-existing aim. Thus, when discussing stochasticity in evolutionary terms it is perhaps better to use evolutionary trajectories, to describe the randomly created paths they move along, instead of "directions."
As an example of convergent evolution we could use the peppered moth and a little imagination because this is purely a hypothetical illustration I am making up on the spot.
The peppered moth is famous for once being a white moth that turned dark during the industrial revolution, when towns and cities grew and pollution killed the light coloured lichens that helped the light morph hide. This meant, rather suddenly by evolutionary time-scales, the white morph of the moth was not very well camouflaged against the dark bark of the trees. This created a positive selection for darker morphs, as such, alleles which made for dark moths would have spread rapidly in the population.
Using a little imagination we can think about two different parts of the world where this same event occurred independently - e.g. Glasgow and Plymouth in Britain which are quite far apart but both grew expansively during the industrial revolution. In both of these locations we would have seen a simultaneous rapid loss of lichens and increase in the dark moth ecomorph. This would have been convergent evolution because it is likely that they were independent of each other, i.e. the dark moth probably did not evolve in Glasgow and then fly itself down to Plymouth.