It is absolutely possible, yes, although more serious tears are accompanied with swelling not too long after, which limits motion. Often people find their knee doesn't feel all that weird after, only for it to worsen later on.
Your doctor isn't 100% sure, and there really is no way of telling unless you go get an MRI, which is what I suggest (if you can). When I tore my ACL (complete tear) and MCL (grade 2-3, almost complete tear) and meniscus, my doctor initially told me that it was probably just a sprain and it would sort itself out. I came back after 2 weeks, he did the usual tests to test for shin movement and spasming calf muscle, yet told me I would be fine a couple more weeks. I asked him to refer me for an MRI (cause you need a referral to get one in the UK), and he told me that wasn't necessary. A couple weeks later, my knee still wasn't stable, although the swelling and stuff and gone down, but by this time I was already on my way back to Canada. Went to my family doctor, he did the same shin-bone and calf-muscle test, told me I could have a partial ACL tear but to rule out a complete tear. I went to a physio (who works with the Canadian alpine team, so he knew his shit), after he performed some tests he told me it's pretty likely I had a fully torn ACL. So I went back to my family doctor, asked him for a referral, then ended up shelling out $500 for a private MRI cause at this point I was pretty fed up (this was nearly 2.5 months after the injury). Results came back and it turned out I had a ACL+MCL+meniscus tear. Needless to say, I was pretty annoyed with the way the doctors just kinda brushed it off with the whole "you'll be fine, come back next week" attitude.
Anyway, wait a couple weeks and if it still isn't better, ask your doc if you can get an MRI (you won't be able to get one now anyway, if there's swelling... you either MRI immediately after or wait until swelling goes down). Better to be completely sure than to fuck around with your knees.