In general
Yes! A simpler example might be a mass in a 2-D plane attached to four springs (one on each side). Assuming the oscillations are small, the motion of the mass can be described as the combination of oscillations in both $x$ and $y$ directions.
Solving your example
In fact, in your specific example, we do see that phenomenon (for small oscillations!).
Assume we have a pendulum where the string is actually a spring. 782
For simplicity, I'll restrict the motion of the pendulum to one plane (ie it can swing in one direction, but is blocked along the other).
The total energy can be written as the sum of kinetic energy (in both the swing direction and radial direction), and the gravitational potential energy and the spring potential energy.
$E(l, \theta)=\frac{1}{2}m\dot{l}^2+\frac{1}{2}ml^2\dot{\theta}^2+\frac{1}{2}k(l-l_0)^2+mg(l_0-l\cos\theta)$
where $l$ is the length of the pendulum spring, $l_0$ the "natural" length of the pendulum spring if you lay it flat on a table, $\theta$ is the angle from vertical, dots indicate time derivatives. I set the $E=0$ reference for the gravitational part as when the pendulum is at the bottom of its motion.
Now let's assume the swinging motion is small: if we Taylor expand the $\cos\theta$, keeping only up to second-order in small $\theta$:
$E(l, \theta)=\frac{1}{2}m\dot{l}^2+\frac{1}{2}ml^2\dot{\theta}^2+\frac{1}{2}k(l-l_0)^2+mg(l_0-l)+\frac{1}{2}mgl\theta^2$
By completing the square, we can combine the third and fourth term, and (ignoring a constant term that appears and just shifts the energy), we find
$E(l, \theta)=\frac{1}{2}m\dot{l}^2+\frac{1}{2}ml^2\dot{\theta}^2+\frac{1}{2}k\left(l-l_0-\frac{mg}{k}\right)^2+\frac{1}{2}mgl\theta^2$
Let $x=l-x_0$ where $x_0=l_0+\frac{mg}{k}$
$E(x, \theta)=\frac{1}{2}m\dot{x}^2+\frac{1}{2}kx^2+\frac{1}{2}m(x+x_0)^2\dot{\theta}^2+\frac{1}{2}mg(x+x_0)\theta^2$
The first two terms look like our usual harmonic oscillator in $x$, oscillating around $x=0$.
We note that, if the swing is small ($\theta$ is small) and the spring movement is small ($x$ is small compared to $x_0$), then $x\dot{\theta}^2$ and $x\theta^2$ have three factors of smallness, and $x^2\dot{\theta}^2$ has four! I'll get rid of the $x^2\dot{\theta}^2$ term since we already killed off some terms with four factors of smallness when expanding $\cos\theta$. Now, rearranging:
$E(x, \theta)=\frac{1}{2}m\dot{x}^2+\frac{1}{2}kx^2+\frac{1}{2}mx_0^2\dot{\theta}^2+\frac{1}{2}mgx_0\theta^2 + P(x, \theta)$
where $P(x, \theta)=mxx_0\dot{\theta}^2+\frac{1}{2}mgx\theta^2$.
Interpreting
Forget about $P$ for a moment, and you see we have two independent harmonic oscillator motions happening at once:
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