Can you have strong values and still act unethically?  Of course. 

Declaring your values is not enough.  Your actions must support your values.  Acting in accordance to your values requires continual attention and thoughtfulness because each situation is unique.  

Being able to think critically about a situation is imperative to living in accordance with your values. 
Critical thinkers:  
  • gather and assess information and then interpret that information to understand a situation
  • test their interpretations against criteria (if it is a decision about ethics the criteria is their values)
  • think openmindedly about their assumptions, and the the consequences and implications of their interpretations and actions.   (these bullets adapted from the Critical Thinking Community website)

Chris McDonald at The Business Ethics Blog, asserts that wrongdoing many times has more to do with lazy thinking than it does with poor values: 

The key to wrongdoing is not primarily that wrongdoers have the wrong values (from which it would follow that ethics classes need to accomplish the difficult, perhaps impossible, task of instilling the right values in just a few short months of instruction). The key to wrongdoing is much more likely to involve faulty ways of thinking about certain behaviours, namely thinking about them in ways that "neutralize" them, morally, effectively exempting the wrongdoer from moral blame. (A simple example is the redescription of theft as "borrowing", or the redescription of stealing from one's employer as "merely taking what I deserve"). The arguments behind such neutralizations are generally fallacious, and fallacies of reasoning are something that can be taught, either in an ethics class or indeed in a first-year Critical Thinking class.
See the entire post here