Four and five term recurrences are found for some special $_3F_2$ and $_4F_3$ type hypergeometric polynomials. As a consequence we shall get the Apéry recurrences for the sums $\displaystyle \sum _{k=0}^n {n\choose k}^2{{n+k}\choose k}^i$ $(i=1,2)$ and some other three term recurrences - like $A(n)G_n+B(n)G_{n-2}+C(n)G_{n-4}=0$ for $\displaystyle G_n=\sum _{k=0}^n{n \choose k}^4(-1)^k$ $(\deg A(n)=\deg B(n)=\deg C(n)=6)$.