I applied for a job a few weeks ago. The application took ages and my experience is spot on for the role. Without outing the job I can confidently say that they wouldn't have had hundreds of applications at all as it was advertised very low key, in a local area and the deadline is long gone.
I'm a bit upset that I have not even had a curt response to say thanks but no thanks.
Definitely follow up! I had a similar situation and the company came back to me after 2 months, when I’d already started another job, to say they wanted to interview me. The hiring manager or HR person could be out on holiday which is delaying responses; you may not have been rejected. And even if you have (internal candidate perhaps?), I don’t think you have anything to lose. I’d probably try to email though, getting through to speak to someone might be easier said than done unless it’s a small company.