It probably is possible to solve problems without active thinking, because thinking itself and creating new ideas is a rather passive process.
Problem-solving is tightly coupled to learning, understanding, and creativity. This is probably represented in the brain with neurons and synapses.
While you sleep, you are probably growing new synapses. Your dreams are also an indicator of what's going on in your mind. And it's a very passive process, as well.
You mentioned something of your personal experience:
When I take a rest after hours of thinking and come back a couple of hours later to solve the problem I do not really notice major differences in my understanding.
The following is conjecture, but let's think about exercise science for a minute:
Here, an adaptive response happens only after a stimulus that's intense enough. You won't actually build more muscle after 1 light workout that barely caused fatigue, you have to exert yourself.
Same might be true for problem-solving, you have to really think hard in your waking hours to give those neurons and synapses a reason to grow while you sleep.
So, as with all things in science, you'd have to quantify your own experience with the subject.
How long did you study? How intensely did you study? How long did you rest before returning to the subject, and how are you measuring your progress from problem towards solution?
There may be some individual variation, but overall this is how you'd study the exact metrics of this particular topic.
The synaptic plasticity and memory hypothesis: encoding, storage and persistence
REM sleep selectively prunes and maintains new synapses in development and learning
Sleep deprivation inhibits adult neurogenesis in the hippocampus by elevating glucocorticoids
Physical Stimulus-Performance-Adaptation : Understanding the physiological relationship
Here's an experiment that shows how particles self-assemble, when electricity is applied to them. Just conjecture, but in the brain similar things are probably also happening, as neurons become charged they might attract particles to form synapses:
Self-Assembling Wires by Standford Complexity Group