Is it possible for a router to "go bad" with time?
Yes.
In general, routers can and do fail. The primary cause of failure for consumer grade equipment is heat stress. Most consumer grade hardware runs far too hot and have respectively poor air cirulation compared to their ventilation needs.
Long-term exposure to heat causes various components to degrade/fail and manifests itself as "intermittent" problems. In general, consumer grade hardware is not as robustly made as commercial or enterprise hardware. But all physical devices are subject to physical effects.
It's not uncommon for consumer grade devices to fail within a few years due to heat or vibration issues. Routers stuck near windows (argh! the sun!), placed on the floor (dust!), or jammed into a bookcase (no air flow) are especially prone to failures. Contrast that with commercial grade devices which are often still working for 10 or more years after their first deployment.
Most cable modems have either an ethernet port or WiFi ability. To isolate the cause of your network problems, you should consider bypassing your router and plugging your PC/laptop directly into the cable modem to see if whether or not you experience the same problems.
Of course, bypassing the router means you bypass the router's firewall protection and NAT abilities so take due precautions on your computer.
A (possibly) a good example of the Second Law of Thermodynamics:
«Any transformation of a thermodynamic system is carried out with increase in entropy including overall entropy of the system and the external environment.»
You wrote:
If I reset the modem and the router, they work fine for a few hours, but the problems start again after a while.
This may be an overheat problem or the overheat is the symptom...
The easiest way to check if the router is Out of Service or near to this inevitable state, you may try with another one temporary (from a friend for example). If this solved the Internet connection problems, you have the answer...
Entry level D-Link Routers, and older d-link routers are not known to be very stable. They were marginal devices years ago, and they are certainly not appropriate for today's general use with many computers and smartphones connected to fast (>15mbps) broadband.
To answer your question specifically, I worked for an ISP that had D-Link products. While some were decent, if unspectacular, they began failing in large bunches 3-5 years after they were put in use.
As a general rule, if a router needs to be restarted more than once a month, junk it.