Generally, No. Even fans, in some cases, were optional.
Heatsinks showed up for later 486 120 and Pentium systems, But thermal paste didn't really become necessary until the faster Pentium II chips.
Thermal throttling behaviour would explain your symptoms, however, Thermal Throttling is relatively recent in terms of CPU capabilities. An overheating Pentium 200 Processor, like in that system, isn't going to underclock itself to lower it's voltage to protect itself. It's just going to fail.
I'm not suggesting your processor is failing, mind you. Your symptoms probably have another cause.