Yes you may have both TrueType and OpenType fonts installed on the same computer! However, you should avoid having two fonts with the same exact name installed, regardless of what format they are.
Adobe Creative Suite applications differentiate between Type 1, TrueType, and OpenType CFF fonts that have the same font name (but obviously not the same font file name), list them separately, and allow one or the other to be selected for particular text objects. However, how the operating system and other applications handle this could be very problematic and our experience in the past in fact confirms this concern. This is exactly why Adobe used different names for the fonts in its library when it created OpenType CFF versions of the fonts (Adobe OpenType CFF fonts typically have “Std” or “Pro” in their names to signify “standard” character sets or expanded, often highly multilingual “professional” character sets).
I personally like to err on the side of paranoia in terms of avoiding fonts with the exact same name but different font types!