Answer

A player cannot place additional data in the lower spaces of computer techs if the upper spaces are not already full. According to the rules:

• "You can place data in the lower space only if the upper space is already full."
  (Page 17, Computer Techs section)

• "You don't have to fill the lower spaces before you can analyze data – the upper spaces are the only ones that have to be filled."
  (Page 17, Computer Techs section)

• "Data is placed from left to right in the upper spaces. Data in the lower spaces do not need to be filled in from left to right; they can be filled in at any point provided that the space above has been covered."
  (Page 17, Computer Techs section)

• "Your data pool can never hold more than 6 data at a time. If you would ever gain data when your data pool is full, you must discard the excess."
  (Page 14, Data Capacity section)

• "Placing data in your computer is a free action. You can perform free actions whenever you want during your turn, but you can't place data when it is not your turn."
  (Page 14, Placing Data section)

How this applies

• If your computer is full (all upper spaces are filled), you cannot place more data in the upper spaces.
• You may place data in the lower spaces of computer techs as long as the upper space above each lower space is already filled.
• If you have data in your data pool and there are eligible lower spaces (with their upper space filled), you can place data there as a free action before analyzing.
• If there are no eligible lower spaces (because their upper spaces are not filled), and your computer is full, you cannot place additional data. Any data you would gain beyond your data pool limit is lost (discarded).

Summary

• You can only place data in lower spaces of computer techs if the upper space above is already filled.
• If your computer is full and there are no eligible lower spaces, you cannot place additional data; any excess data is lost.
• This is supported by the rules on pages 14 and 17.

Source:
Page 14 (Placing Data, Data Capacity), Page 17 (Computer Techs)

Source: Core Rulebook (pp. 14, 16, 17, 26, 28)