Not usually, in my experience, and the reason why is that they are usually judged by objective standards with an objective set of criteria and you're really only going to win the kind of contest with those kinds of guidelines if you tailor your writing to what the judges are going to be looking for. With this in mind, when you submit to that kind of a contest, are you really submitting your own work? If not, then clearly whatever the judges say isn't a reflection of your words, it's a reflection of your words tailored to a specific format. It's like the difference between writing an editorial or opinion piece, which you have a free hand with and can say mostly anything because it's your opinion, and a thesis paper where you're constrained to APA format and have to stick to the topic that your professor wants to see you write.
Another other issue is that my appreciation of writing, and think many other's also, is based on subjectivity and my own appreciation, without much consideration for objective guidelines or prescriptive rules, whereas a judge should be looking for all of the elements specified in the rules for the contest and may appreciate your story, but won't give that as feedback necessarily if they feel you skirted the edge of one of the rules too closely.
Finally, all of the contests I've seen in the past few years look to me like scams where they are trying to get as much free writing as possible to publish without paying for it.
That said, if you can find a contest or a set of submission guidelines that are a fit for your writing without changing your style at all, or with minimal changes, then I'd say yes, it can be a good way to get an idea of the quality of your writing.
If it were me, however, rather than look for a contest I'd be looking for publications like magazines, anthologies, blogs, and other small distribution channels that are looking for submissions, and trying to get what's called a beta reader to give me feedback. The publications I've seen tend to pay 4-8 cents a word and tend to want 2k-6k words, so you'll be getting paid to be an author, and in general I've seen publications submission guidelines be much looser than contest entry rules so you'll have a more of a free hand to express your words.
YMMV, I hope this helps.