More importantly, you cannot solve problems or let go of problems if you don't allow yourself time to think about them. This is a concept ignored by our culture which overvalues doing more rather than thinking and believes answers are in in the palm of your hand rather than in your head.
Suppressing negative feelings only gives the feelings more power which can lead to intrusive thoughts which makes people try to stay busier to keep the thoughts away. The cognitive strain of evading emotions underlies a range of psychological difficulties including, depression, anxiety, panic disorder and addictions/addictive behaviors and can be associated with a number of physical ailments like skin problems, arthma, stomach and headaches to name a few.
Other scientific studies suggest that not having time to reflect impairs you ability to empathize with others and yourself. Additionally, researchers have found an idle mind is critical to creativity. The company 'Google' offers it's employees courses called "Search inside Yourself" and "Neural Self-Hacking". These courses include mindfulness, meditation with the goal to recognize and accept inner thoughts and feelings rather then ignore or repress them.
Hard as they may seem negative feelings are a part of everyone's life. Emotions are important communications to yourself about yourself. While busy-ness may help in the short term to evade sadness, it also limits one's ability to experience joy!