Can connections between neurons be weakened?
Short answer
Yes, neural connections can be weakened.  

Background
Chemical connections between neurons (synapses) can certainly be strengthened or weakened, dependent on their input. Single synapses can undergo long term potentiation (LTP) or long term depression (LTD). As a typical example there are hippocampal neurons. High-frequency (tetanic) electrical stimulation in hippocampal neurons can result in long-term potentiation (strengthening) of synaptic connections in the hippocampus Moreover, when a synapse receives synchronous input from multiple incoming axons they can potentiate each other's responses (Purves et al, 2001). In contrast, low-frequency stimulation and  prolonged inactivity of synapses in the hippocampus can weaken them (Bear &amp; Abraham, 1996).  

The connection between learning, let alone good and bad as you ask for in your question, is far from clear. However, LTP and LTD do provide plausible mechanisms that could underpin these processes at the cellular level (Purves et al, 2001).

References
- Bear &amp; Abraham, Ann Rev Neurosci (1996); 19: 437-62
- Purves et al., eds. Neuroscience, 2nd ed., Sunderland (MA): Sinauer Associates; 2001