I will first show that the general answer is no (in fact one should expect the opposite even in the graded case) by the Pigeonhole principle. Then we shall construct some concrete examples via an old result of Tate.
Let $(A,k)$ be a local artin algebra. We write $A=R/I$ as a (minimal) quotient of a regular local ring $(R,k)$. Note that $\dim R$ is bounded above by $d_A-1$.
Claim 1: Fixing $d_A$, then the set of possible sequences $\{c_i\}_{i\geq 0}$ is countable.
Proof: this follows from an inequality proved by Serre:
$$\sum c_it^i \ll \frac{(1+t)^{\dim R}}{1-\sum_{i>0}\dim_k Tor_i^R(k,A) t^{i+1}} $$
(where $\ll$ means that the LHS is term by term dominated by the RHS). For references, look up "Golod rings", this is such a popular subject even I had a paper about it.
Now, to finish the proof, note that the denominator of RHS is a polynomial of degree $\dim R$, and each $\dim_k Tor_i^R(k,A)$ is bounded above by $d_A\dim_k Tor_i^R(k,k)$ as $A$ has a filtration by $d_A$ copies of $k$ and $\dim Tor_i(k,-)$ is subadditive on short exact sequences. Finally, $\dim_k Tor_i^R(k,k) = \binom{\dim R}{i}$.
Claim 2: Fixing $d_A$, and assume $k$ is algebraically closed, the set of isoclasses of $A$, even assuming $A$ standard graded, can be uncountable.
Proof: this comes from standard facts that the set of isoclasses of $A$ with a fixed Hilbert function are parametrized by Hilbert schemes.
Putting together Claim 1 and 2 shows that there are sequences with uncountably many $A$s.
Example: For a concrete example, a result by Tate (mentioned in Eisenbud's paper on complete intersection) implies that when $A$ is a complete intersection, the $\{c_i\}$ only depends on $\dim R$, so even two general quadrics in $k[x,y]$ would give infinitely many examples.
