If you are partically paying the rent, which the landlord accepts - can they still evict you?

Yes. The landlord can proceed with filing for eviction as long as there is any rent due.

Do you mean partially paying rent? When I first got an apartment we had this agreement with the landlord cause money was tight, we paid the rent in 2 or 3 installments but it was always paid before the 1st of the month. For example I would pay 1/2 June 14th and 1/2 June 28th for July's rent.

You need to make sure you are both on the same page, they are probably okay with accepting rent in installments but expect to have it in full before the 1st of each month.