Is it possible ?
Rather obviously yes. Let $A$ be the algebra over the field $K$ generated by elements $a_1,a_2,\ldots,$ with $a_i$ in dimension $i$ and with $a_ia_j=0$ for all $i$ and $j$. This is an incredibly uninteresting example, but since each graded piece is onedimensional, its Poincare series is $\sum_{n=0}^\infty t^n=1/(1t)$. 

