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Using mutable keys in data collections such as HashMap or HashSet in Java can lead to serious problems. The main reason is that keys should be immutable so that the hash code and equality remain constant.

If a key changes after being added to the collection:

• The hash code of the key may change, and the collection will not be able to find the element by this key.
• This will result in the 'loss' of the element: it will remain in the structure but will be inaccessible.
• The integrity of data and the correct functioning of the collection will be compromised.

Example:

Map<List<String>, String> map = new HashMap<>();
List<String> key = new ArrayList<>();
key.add("initial");
map.put(key, "value");

// Change the key after adding
key.add("changed");

// Attempting to get the value by key will not work
String val = map.get(key); // null

Therefore, it is recommended to use immutable objects as keys, such as strings, numbers, or specially created immutable classes.