The word постепенно
indeed means gradually
but I would also be focused here on the verb подобралась
. Because it is not a common use case of this word.
Strictly saying this word should be used when a speaker described a situation where some collection was chosen from a variety of collections whilst (s)he needed only one specific collection fitting to his/her particular purpose. Like in this example:
Код к электронному замку подобрался
не сразу
or
Она была высокого роста, но, тем не менее, партнёр по танцам ей постепенно подобрался
Another, much far use case of this word can be found it this example
Этот ранненый медвеженок подобрался
нашей группой во время экспедиции и был послан в зоопарк. Here in in the meaning of "to be taken"
or
Зверь подобрался
к своей жертве на расстояние прыжка. Here in the meaning of "to get closer"
But that's another story :)
However, if the speaker's intention was just expressing the point that the collection had been completed with time delays or just not quickly, then the word should be changed on собралась
:
Коллекция собралась постепенно
At the same time I believe that the certain sentence (as it is in the question) can be well and even equal to the latter in old fashion language, probably.
Another point proving this statement could be the next consideration. In modern Russian language this word подбираться
usually requires a proper context with the conditions and requirements on for which purpose the main object, like the collection
should fit, for example:
Я собирался сдавать вступительный экзамен по литературе и, к счастью, к этому моменту у меня подобралась
коллекция великолепных эссе.
or
Нашему отделу дали новый проект, но шансов его успешно выполнить очень мало, так как коллектив у нас подобрался
не по этому профилю.
As can be seen the word постепенно
is not the one which is used normally with подобрался
.
To sum up, I would say it's definitely possible in the expressing even simple idea of time delays during collecting - native speakers will not hear a mistake in it, but at the same time that would make them expecting you are either extending a context or being originally from a previous epoch ;) Hope it helps.