I have a 1 year old cat and she gave birth to four kittens five weeks ago, so she is still breastfeeding them. My question would be: is it possible that she is pregnant again (she can go out any time)?
Theoretically yes. Cats are pregnant for just over 60 days and generally goes on heat every 3 weeks or so. It is known that the heat cycle can start very shortly after giving birth, meaning that she will show interest and be susceptible to tom cats again.
If you want a definitive answer on whether she IS pregnant, I would advise a consultation with your family veterinarian who might be able to tell by palpation (not always) or give you options to do a sonar or blood tests.
In this case, if I were the veterinarian, I would strongly recommend spaying her to prevent (unwanted) offspring.
Generally, adult female house cats are pregnant, nursing, in heat, or the one to two weeks between heat cycles. If she is currently nursing, she is not likely to have gone back into heat and be impregnated again. (One of the reasons adult males of various species, including felines, kill young is so the female will go back into estrus and they can mate with them.)
Given that she gave birth about 5 weeks ago and is still nursing, it is possible, but not probable, that she is pregnant.
It is possible, yes that she may be pregnant again. Most cats do not come into heat again until after they have finished nursing their kittens, but in rare cases a queen may enter heat within the first few days of giving birth.
If your cat was housebound towards the end of her pregnancy, and you can confidently vouch that she was not around an entire male immediately after giving birth, then the chances of her being pregnant again are pretty much none.
Generally, whilst nursing a queen will not enter a season, but if she has been going outside since parturition, then it is possible, but unlikely that she may have entered into another season and found a male to mate with.