To determine whether a triangle can exist with a given area of $12 \, \text{cm}^2$ and a perimeter of $12 \, \text{cm}$, we employ the following steps:

The area $A$ of a triangle with sides $a, b, c$ can be calculated using Heron's formula:
$$A = \sqrt{s(s-a)(s-b)(s-c)},$$
where $s$ is the semi-perimeter of the triangle, given by $s = \frac{a+b+c}{2}$.

Given a perimeter of $12 \, \text{cm}$, we have:
$$a + b + c = 12.$$
Thus, the semi-perimeter is:
$$s = \frac{12}{2} = 6.$$

According to Heron's formula, the area condition becomes:
$$12 = \sqrt{6(6-a)(6-b)(6-c)}.$$

To find possible integer side lengths, let's assume $a \leq b \leq c$. From the triangle inequality, it follows:

1. $a + b > c$
2. $b + c > a$
3. $c + a > b$

Given these constraints and the perimeter condition $a + b + c = 12$, we consider possible combinations for $(a, b, c)$ where $a, b, c$ are positive integers. Testing feasible solutions by checking:
$$12 = \sqrt{6(6-a)(6-b)(6-(12-a-b))}.$$

We test different integer values for $a, b, c$ that satisfy the equations and maintain the perimeter at $12$, but find no suitable set of values that yields an area of $12 \, \text{cm}^2$.

Furthermore, the triangle inequality can be more deeply analyzed within the fixed perimeter context to observe that no such triangle exists with both required area and perimeter due to conflicting conditions typically emerging from equal distribution of side lengths to maintain the perimeter constraint while achieving a higher area.

Thus, based on calculations and geometric reasoning, there is no triangle that satisfies both the given area and perimeter constraints.

$$\boxed{\text{No}}$$