The only situations I've really run across problems with color is with labeling or tagging, if colors play a critical role in distinguishing the tags. This was particularly bad in Trello before they introduced a color-blind mode. I actually dislike the way they handled it. Trello only allows 6 predefined colors for labels, and the colors they picked were particularly bad for those with red-green colorblindness (I could barely distinguish between the blue and purple, and could distinguish the red and orange side-by-side but not identify in isolation); they then introduced a color blind friendly mode that adds unique patterns to the most problematic colors (blue, orange, green). What bothers me about this is that it would have been extremely easy to just pick 6 colors that weren't problematic (red, dark blue, light blue, yellow, black; there, that took me under a minute, and the least you could do is consult a representative of 8% of your users for 30 seconds). Once they realized the colors they'd chosen were a problem for red-green color blind users, I understand why they introduced a new mode instead of changing the colors for everyone, since that would disrupt and confuse many people ("I don't know where I put the card, I just made it purple, and now there's no purple!"), but the problem could have been much more elegantly solved with a bit more forethought.
Summary: color blindness is only a handicap in situations where identifying color is critical. Situations like this aren't that common, but there are cases, and when you're going to put significant thought into picking a color palette it just comes off as lazy and irresponsible (unless there really isn't a good solution—e.g., picking a palette of 50 colors).