I have a few thoughts with varying level of risk.
Suggestion #1 - Minimal Risk
Go to www.icloud.com and see if there's anything in there that you can delete. This is going to sound weird, but after you've made more space in iCloud, go into the Calendar app. At the bottom middle, you will see "Calendars." Touch that. It will bring up a list of your calendars. Swipe down on that screen. I've found that Contacts and Calendars are linked and by refreshing one, you refresh them both.
Suggestion #2 - Somewhat Risky
Go into Settings > iCloud. Slide the Contacts option to OFF. It will ask you if you want to keep your Contacts on your device. Save them on your device. Restart your phone. Go back into Settings > iCloud and switch Contacts back ON. You'll get a message asking if you want to merge your contacts, say yes. It should merge what's saved on iCloud and what's on your iPhone. I say, "somewhat risky," because this is an ideal situation. If there is something weird going on with your iOS or Contacts app, the outcome could be unpredictable.
Suggestion #3 - Somewhat Risky
Go into Settings > iCloud. Sign out of iCloud. Sign back in. I recently did this and had no problems with my contacts, but again... if you have an issue with your iOS or the Contacts app, the results could be unpredictable.