The major problems occur on Kilimanjaro, where due to the costs being based on the number of days you are on the mountain, many organised groups ascend far faster than the recommended 300m per day. It is possible to spend time on some of the adjacent mountains to help acclimatise before tackling the main summit.
Some companies are starting to offer itineraries which take account of this. It is worth spending a bit extra to stay safe and achieve your goals. Find out from the trekking company how many people did and how many didn’t reach the top (or Gilmans Point) on their last few trips. Ask how many days they were trekking. A safe ascent should take 8 to 10 days.
An experienced British Mountain guide with training in mountain medicine was leading a group up Mt Kilimanjaro (5,895m). They came across a very scared group with a collapsed 17-year-old boy. The guide injected dexamethasone at the same time as commencing a rapid and life saving descent. On arrival at a hut shelter he used his satellite phone to contact a UK climbing doctor who advised further descent through the night. Two days later the boy was fit and well. Had anyone in the group known a little more about altitude illness they may have been able to avoid the emergency. Sadly they all learned some lessons the hard way.