Most people will be familiar with the idea that naturalistic spelling can be tricky. Some examples of natlangs with annoying spelling that come to mind are English ("ough" can be pronounced /ʌf/, /ɔf/, /ʌp/, /ow/, /u/, or /o/), French ("Qu'est-ce qu'il mangent ?" /kɛs kɪl mãʒ/), and Modern Greek (/i/ can be spelled "η", "ι", "ει", "υ", "οι", or "υι"). But how bad can it get? Are completely silent words (that is, written but unpronounced) words attested in any language? I'm aware that Irish mutations developed from completely lost particles, but I don't think these particles are still written as separate words.
Dropping words happen (the ne of negation notoriously drops pretty systematically in spoken French) and commonly a that, as Gregory points out. However, I believer a true "silent word" is a nonsensical concept.
For starters, Silent letters are historical artifacts of a writing system. They are not a feature of the spoken language, so right there, a "silent word" cannot be something that happens in the spoken language unless the words in question are composed of phonemes or features that are merely difficult to hear from locutors of a different language. Still not silent in their native language. If a writer writes down a word, they typically do it with some sort of communicative intent (even if not consciously so).
Second, in the spoken language, something that isn't pronounced in one way or another literally doesn't exist by definition (we can argue about stuff like sandhi and the side effects of sound change, but that's beyond the scope of this discussion IMO).
So unless your language has a need to transcribe something that is literally not said (and we usually have punctuation and emoticons for that), then a "silent word" is, in my opinion, a self-contradicting concept.
In terms of spelling opacity, there's a continuum between exactly mirroring speech at the phonetic or phonological level and writing in a different language. Historical spelling normally refers to writing segments, especially consonants in clusters or inflectional morphology, that have been lost over time. I think adding back lost morphology or undoing sound changes in a much more common kind of historical spelling than adding back lost syntax.
There are some historical changes that cause words to be dropped without turning into affixes, though. For instance, a language can develop a null copula construction like Russian did for the present tense.
It would not be too much of a stretch to imagine a language where the null copula is a fairly recent innovation, and therefore the copula is usually written despite not being spoken. This may have been the case for Russian at some point in time.
I don't know of any examples, but you might create a language where things such as articles or measure words existed at one point but were lost. I think spontaneously losing articles or measure words is uncommon, so it would probably be a contact-induced change.
I'll start by saying I'm not an expert, only an enthusiast.
But I believe this could be entirely possible. Written languages such as Egyptian never had punctuation. So they would not be capable of using punctuation to indicate certain moments in speech indicated by them. I could see how a similar constructed language would be capable of having a silent word being used that represents these punctuation moments.
For example, a sentence where you want to indicate a emphatic pause could simply use a word that might not have a normal use, like combining the words for "wait" and "moment" into a new word that when written symbolize the spoken need for a pause of emphasis.
The archaeological site L'Anse aux Meadows is pronounced (in English) /lænsi mɛdoʊz/, i.e. the aux is silent, if random internet sources can be believed. Of course, it is not quite a typical example - a proper name, derived from another language, and English speakers unaware of the local pronunciation are likely to read differently. But that's the fate of the English orthography.