Changing from a copyleft licence (e.g. GPL) to a weak copyleft (e.g. LGPL) or to a permissive licence (e.g. Apache License) is almost never a good idea.
Using a weak copyleft or a permissive licence can be a good idea if:
The Ogg media suite is an example of when it's good to use a permissive licence.
A seemingly bad change:
Bradley Kuhn says QT's GPL -> LGPL switch was a good move: LGPL'ing of Qt Will Encourage More Software Freedom
status.net was AGPL v3, but the developers chose Apache License 2.0 for the next generation replacement, pump.io. The lead developer, Evan Prodromou says the change is because the software fits the three criteria mentioned above.
Bradley Kuhn, in his Fosdem 2014 talk, said this was a good idea (or maybe he said "probably" a good idea), because pump.io aims to establish a set of web protocols. Listen to the audio of his talk for more detail (it will be online soon, or maybe is already).
LibreOffice (LO) was originally LGPLv3+ but has switched to MPL2. More precisely, it will be dual-licensed under both, but distributors can only be held to the weaker of the two sets of requirements, which is MPL2.
Is this justified? Maybe.
Most of the functionality in LO was already made available under permissive and proprietary licences by Sun Microsystems and later by Apache Open Office (AOO). This puts LO in a weak position because if they ask third-party projects to abide by the weak copyleft requirements of the LGPL, the third-parties have the option of saying no and working with AOO instead. (More info about this race to the bottom can be found in the replies to this comment: [1].)
"there are plenty of other C libraries; using the GPL for ours would have driven proprietary software developers to use another" as explained in Why you shouldn't use the Lesser GPL for your next library.
An example of crafting narrow GPL exceptions to address particular use cases rather than moving to a weaker copyleft or permissive license wholesale; in depth explanations by Bradley Kuhn and in the GCC runtime exception FAQ and rationale.