For dating fossils as far back as the late Precambrian a new dating method called the rhenium - osmium method has recently been used, and is much more accurate for that period than previous methods, being capable of reliably dating fossils a billion years old. Previous inaccurate datings were rendered obsolete.
Fossils found in limestone cave accretions such as Raymond Dart's Taung child (a gracile australopithecine from S. Africa) are notoriously difficult to date because of the lack of stratigraphy and suitable radioactive elements.
Fossil morphology must also be considered to determine if you are really looking at a new species or just a special case of a previously identified species. Sometimes, evolutionary approaches can be applied to help place the specimen into a timeline.
Figuring out the temporal extent of a species takes more time and multiple discoveries of samples of that species to better constrain that extent.
A combination of approaches and techniques can help paleontologists be detectives in determining the age of specimen. Like other sciences, geology is social in the sense that it relies on the culmination of the work of many, with multiple lines of complementary pieces of information, to come to a conclusive age.
Here are some great articles related to your question:
Here is an example of an "undatable" fossil, where the age could "only" be constrained by 2 million years:
Bad Luck again. There is one other circumstance, rarely you have reworked material, that is fossils that where exposed by erosion in the distance past then reburied, This is exceedingly rare, and always leaves its mark on the fossil, in that case we know the fossil is in the wrong place but may not know exactly where and thus when it comes from originally. Usually you can identify by finding similar fossils, but rarely the fossil so badly degraded identification is impossible.
http://preparation.paleo.amnh.org/35/techniques-in-the-field
To answer your question, "...surely there have been cases where we have found new fossils that we have no way of knowing the age?", the short answer is yes.
Prior to modern radiometric dating in the early 1900s, which placed firm constraints on age, you could say that all fossils found until that point were of unknown age. Sure, you could establish depositional constraints: this fossil is older because it lies under that one, but no one knew how old, relative to the age of the earth, it was, or how old relative to each other they were.
For a long time, at least throughout Europe, the Bible was the chief source of all information and natural philosophers assumed fossils were victims of The Great Flood. If you spent all of your time studying the bible for clues for when God created everything, you'd back-calculate that nothing on earth could be older than about 4,000 years old. They were derided as "Antediluvialists" by later generations of thinkers for whom the concept of deep time was a very real possibility. Even Lord Kelvin, the smartest man in the world at the time, couldn't understand why fossils needed to be so old.
Discounting Tinder, it's impossible to find an undatable fossil, even those that have survived metamorphism.
An analogous situation existed for European history. Prior to carbon dating, it was believed that all of European culture was derived from the Greeks, as it was (by circular reasoning) the "oldest" that anyone could find because all artifacts found throughout Europe had to be derivations.
When carbon dating revealed European cultures pre-dating the Greeks by thousands of years, an entire school of thought that lasted centuries, vanished literally overnight.